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    }
    .operation-tag {
      font-size: 11px;
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    .rule-formula {
      font-size: 17px;
      font-weight: 600;
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      text-align: center;
      padding: 2px 0;
      display: flex;
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      justify-content: center;
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    }
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      align-items: center;
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      gap: 6px;
      min-height: 52px;
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    }
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      background: #ffffff;
      border-color: rgba(34, 183, 255, 0.5);
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      font-size: 10px;
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      text-transform: uppercase;
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    }
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      font-style: normal;
      display: inline-flex;
      align-items: center;
      line-height: 1.2;
    }
    .math i {
      font-style: italic;
      padding: 0 1px;
    }
    .math .sym {
      margin: 0 3px;
      font-family: system-ui, -apple-system, sans-serif;
      font-weight: 400;
    }
    .math sup {
      font-size: 0.75em;
      line-height: 0;
      position: relative;
      vertical-align: baseline;
      top: -0.55em;
      margin-left: 0.5px;
    }
    .frac {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      padding: 0 1px;
      font-size: 0.85em;
      line-height: 1.1;
      margin: 0 1.5px;
    }
    .frac.large {
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    }
    .frac-sup {
      font-size: 0.72em;
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      line-height: 1;
      padding: 0 1px;
    }
    .num {
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      padding-bottom: 1px;
      display: block;
    }
    .den {
      padding-top: 1px;
      display: block;
    }
    .sqrt {
      display: inline-flex;
      align-items: center;
      margin-left: 1px;
    }
    .sqrt-radical {
      font-size: 1.15em;
      line-height: 0.8;
      margin-right: -1px;
      transform: translateY(1px);
    }
    .sqrt-content {
      border-top: 1.15px solid currentColor;
      padding: 0 2px;
      line-height: 1;
    }
  </style>
</head>
<body>
  <div class="grid" role="region" aria-label="Index Laws Reference">
    
    <div class="law-card active" onclick="selectCard(this)" tabindex="0" role="button" aria-pressed="true">
      <div class="card-header">
        <span class="badge">Product Rule</span>
        <span class="operation-tag">Add Powers</span>
      </div>
      <div class="rule-formula">
        <span class="math"><i>x</i><sup><i>a</i></sup> <span class="sym">&sdot;</span> <i>x</i><sup><i>b</i></sup> <span class="sym">=</span> <i>x</i><sup><i>a</i>+<i>b</i></sup></span>
      </div>
      <div class="example-box">
        <span class="example-label">Example</span>
        <div class="example-math">
          <span class="math"><i>x</i><sup><span class="frac-sup"><span class="num">1</span><span class="den">2</span></span></sup> <span class="sym">&sdot;</span> <i>x</i><sup><span class="frac-sup"><span class="num">1</span><span class="den">3</span></span></sup> <span class="sym">=</span> <i>x</i><sup><span class="frac-sup"><span class="num">5</span><span class="den">6</span></span></sup></span>
        </div>
      </div>
      <div class="step-detail">
        <span class="math"><span class="frac"><span class="num">1</span><span class="den">2</span></span> <span class="sym">+</span> <span class="frac"><span class="num">1</span><span class="den">3</span></span> <span class="sym">=</span> <span class="frac"><span class="num">3</span><span class="den">6</span></span> <span class="sym">+</span> <span class="frac"><span class="num">2</span><span class="den">6</span></span> <span class="sym">=</span> <span class="frac"><span class="num">5</span><span class="den">6</span></span></span>
      </div>
    </div>

    
    <div class="law-card" onclick="selectCard(this)" tabindex="0" role="button" aria-pressed="false">
      <div class="card-header">
        <span class="badge">Quotient Rule</span>
        <span class="operation-tag">Subtract Powers</span>
      </div>
      <div class="rule-formula">
        <span class="math"><span class="frac large"><span class="num"><i>x</i><sup><i>a</i></sup></span><span class="den"><i>x</i><sup><i>b</i></sup></span></span> <span class="sym">=</span> <i>x</i><sup><i>a</i>&minus;<i>b</i></sup></span>
      </div>
      <div class="example-box">
        <span class="example-label">Example</span>
        <div class="example-math">
          <span class="math"><span class="frac large"><span class="num"><i>x</i><sup><span class="frac-sup"><span class="num">3</span><span class="den">4</span></span></sup></span><span class="den"><i>x</i><sup><span class="frac-sup"><span class="num">1</span><span class="den">4</span></span></sup></span></span> <span class="sym">=</span> <i>x</i><sup><span class="frac-sup"><span class="num">1</span><span class="den">2</span></span></sup></span>
        </div>
      </div>
      <div class="step-detail">
        <span class="math"><span class="frac"><span class="num">3</span><span class="den">4</span></span> <span class="sym">&minus;</span> <span class="frac"><span class="num">1</span><span class="den">4</span></span> <span class="sym">=</span> <span class="frac"><span class="num">2</span><span class="den">4</span></span> <span class="sym">=</span> <span class="frac"><span class="num">1</span><span class="den">2</span></span></span>
      </div>
    </div>

    
    <div class="law-card" onclick="selectCard(this)" tabindex="0" role="button" aria-pressed="false">
      <div class="card-header">
        <span class="badge">Power Rule</span>
        <span class="operation-tag">Multiply Powers</span>
      </div>
      <div class="rule-formula">
        <span class="math">(<i>x</i><sup><i>a</i></sup>)<sup><i>b</i></sup> <span class="sym">=</span> <i>x</i><sup><i>ab</i></sup></span>
      </div>
      <div class="example-box">
        <span class="example-label">Example</span>
        <div class="example-math">
          <span class="math">(<i>x</i><sup><span class="frac-sup"><span class="num">2</span><span class="den">3</span></span></sup>)<sup>6</sup> <span class="sym">=</span> <i>x</i><sup>4</sup></span>
        </div>
      </div>
      <div class="step-detail">
        <span class="math"><span class="frac"><span class="num">2</span><span class="den">3</span></span> <span class="sym">&times;</span> 6 <span class="sym">=</span> <span class="frac"><span class="num">12</span><span class="den">3</span></span> <span class="sym">=</span> 4</span>
      </div>
    </div>

    
    <div class="law-card" onclick="selectCard(this)" tabindex="0" role="button" aria-pressed="false">
      <div class="card-header">
        <span class="badge">Negative Index</span>
        <span class="operation-tag">Reciprocal</span>
      </div>
      <div class="rule-formula">
        <span class="math"><i>x</i><sup>&minus;<i>a</i></sup> <span class="sym">=</span> <span class="frac large"><span class="num">1</span><span class="den"><i>x</i><sup><i>a</i></sup></span></span></span>
      </div>
      <div class="example-box">
        <span class="example-label">Example</span>
        <div class="example-math">
          <span class="math"><i>x</i><sup>&minus;<span class="frac-sup"><span class="num">1</span><span class="den">2</span></span></sup> <span class="sym">=</span> <span class="frac large"><span class="num">1</span><span class="den"><span class="sqrt"><span class="sqrt-radical">&radic;</span><span class="sqrt-content"><i>x</i></span></span></span></span></span>
        </div>
      </div>
      <div class="step-detail">
        <span class="math"><i>x</i><sup>&minus;<span class="frac-sup"><span class="num">1</span><span class="den">2</span></span></sup> <span class="sym">=</span> <span class="frac"><span class="num">1</span><span class="den"><i>x</i><sup><span class="frac-sup"><span class="num">1</span><span class="den">2</span></span></sup></span></span> <span class="sym">=</span> <span class="frac"><span class="num">1</span><span class="den"><span class="sqrt"><span class="sqrt-radical">&radic;</span><span class="sqrt-content"><i>x</i></span></span></span></span></span>
      </div>
    </div>
  </div>

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Simplifying Algebraic Expressions with Rational Exponents and Surds

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<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Simplifying Rational Exponents</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, "Segoe UI", Roboto, sans-serif;
      color: #1e293b;
      background: transparent;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      max-width: 720px;
      margin: 0 auto;
    }

    /* Controls bar */
    .toolbar {
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 8px;
      flex-wrap: wrap;
    }
    .step-pills {
      display: flex;
      gap: 6px;
      flex-wrap: wrap;
    }
    .pill-btn {
      border: 1px solid #cbd5e1;
      background: #f8fafc;
      color: #475569;
      font-size: 11px;
      font-weight: 700;
      padding: 4px 10px;
      border-radius: 20px;
      cursor: pointer;
      transition: background-color 0.2s ease, border-color 0.2s ease, color 0.2s ease;
    }
    .pill-btn:hover {
      background: #e2e8f0;
    }
    .pill-btn.active {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
    }
    .ctrl-btn {
      border: 1px solid #cbd5e1;
      background: #ffffff;
      color: #1e293b;
      font-size: 12px;
      padding: 4px 10px;
      border-radius: 6px;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
      font-weight: 600;
      transition: background-color 0.2s ease;
    }
    .ctrl-btn:hover {
      background: #f1f5f9;
    }

    /* Pipeline Container */
    .pipeline {
      display: grid;
      grid-template-columns: 1fr;
      gap: 8px;
      position: relative;
    }

    @media (min-width: 540px) {
      .pipeline {
        grid-template-columns: 1fr 1fr;
        gap: 12px 10px;
      }
    }

    .card-wrapper {
      display: flex;
      flex-direction: column;
      position: relative;
    }

    .step-card {
      background: #f8fafc;
      border: 1.5px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px 12px;
      display: flex;
      flex-direction: column;
      gap: 6px;
      cursor: pointer;
      transition: border-color 0.2s ease, background-color 0.2s ease, box-shadow 0.2s ease;
      height: 100%;
    }
    .step-card:hover {
      border-color: #94a3b8;
    }
    .step-card.active {
      background: #f0f9ff;
      border-color: #0284c7;
      box-shadow: 0 0 0 1px #0284c7, 0 3px 8px rgba(2, 132, 199, 0.15);
    }

    .card-header {
      display: flex;
      align-items: center;
      justify-content: space-between;
    }
    .badge {
      font-size: 11px;
      font-weight: 800;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      color: #64748b;
      background: #e2e8f0;
      padding: 2px 7px;
      border-radius: 12px;
    }
    .step-card.active .badge {
      background: #0284c7;
      color: #ffffff;
    }
    .flow-tag {
      font-size: 11px;
      font-weight: 600;
      color: #64748b;
    }
    .step-card.active .flow-tag {
      color: #0369a1;
      font-weight: 700;
    }

    .card-title {
      font-size: 12px;
      font-weight: 600;
      color: #475569;
    }
    .step-card.active .card-title {
      color: #0f172a;
    }

    /* Math expression container */
    .math-content {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 6px;
      padding: 8px 6px;
      display: flex;
      align-items: center;
      justify-content: center;
      min-height: 48px;
      font-family: "Cambria Math", "Times New Roman", "DejaVu Serif", serif;
      font-size: 14px;
      color: #0f172a;
      overflow-x: auto;
      white-space: nowrap;
    }
    .step-card.active .math-content {
      border-color: #bae6fd;
      background: #ffffff;
    }

    /* Connector arrow styling */
    .pipeline-connector {
      display: flex;
      align-items: center;
      justify-content: center;
      color: #0284c7;
      padding: 3px 0;
    }
    @media (min-width: 540px) {
      .pipeline-connector {
        display: none;
      }
    }
    
    .connector-badge {
      display: inline-flex;
      align-items: center;
      justify-content: center;
      width: 22px;
      height: 22px;
      border-radius: 50%;
      background: #e0f2fe;
      border: 1.5px solid #38bdf8;
      color: #0284c7;
      font-weight: 800;
      font-size: 12px;
    }

    /* Desktop arrow overlays between grid items */
    .step-arrow-tag {
      display: inline-flex;
      align-items: center;
      gap: 3px;
      font-size: 11px;
      color: #0284c7;
      font-weight: 700;
    }

    /* Math layout components */
    .m-frac {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      padding: 0 3px;
      line-height: 1.1;
    }
    .m-num {
      border-bottom: 1px solid currentColor;
      padding-bottom: 2px;
      margin-bottom: 1px;
    }
    .m-den {
      padding-top: 1px;
    }
    .m-rad {
      display: inline-flex;
      align-items: center;
      position: relative;
    }
    .m-rad-deg {
      font-size: 0.65em;
      position: relative;
      top: -0.4em;
      margin-right: -2px;
      font-weight: bold;
    }
    .m-rad-sign {
      font-size: 1.25em;
      margin-right: 1px;
    }
    .m-rad-body {
      border-top: 1px solid currentColor;
      padding-top: 1px;
      margin-left: -1px;
    }
    .m-sup {
      font-size: 0.72em;
      vertical-align: super;
      line-height: 0;
    }
    .m-eq {
      margin: 0 6px;
      color: #64748b;
      font-weight: normal;
    }
    .c-coef { color: #b45309; font-weight: 600; }
    .c-x { color: #1d4ed8; font-weight: 600; }
    .c-y { color: #047857; font-weight: 600; }
    .c-z { color: #7e22ce; font-weight: 600; }
  </style>
</head>
<body>

  
  <div class="toolbar">
    <div class="step-pills">
      <button class="pill-btn active" data-step="0">1. Convert Radical</button>
      <button class="pill-btn" data-step="1">2. Distribute Exponents</button>
      <button class="pill-btn" data-step="2">3. Group Bases</button>
      <button class="pill-btn" data-step="3">4. Simplify Result</button>
    </div>
    <button class="ctrl-btn" id="playBtn" title="Play / Pause Auto Cycle">
      <span id="playIcon">⏸ Pause</span>
    </button>
  </div>

  
  <div class="pipeline">
    
    
    <div class="card-wrapper">
      <div class="step-card active" data-step="0">
        <div class="card-header">
          <span class="badge">Step 1</span>
          <span class="flow-tag">Radical &rarr; Power <sup>1/3</sup></span>
        </div>
        <div class="card-title">Convert root to fractional exponent:</div>
        <div class="math-content">
          <span class="m-frac">
            <span class="m-num">
              <span class="m-rad">
                <span class="m-rad-deg">3</span><span class="m-rad-sign">&radic;</span><span class="m-rad-body"><span class="c-coef">27</span><span class="c-x">x</span><sup class="m-sup">6</sup><span class="c-y">y</span><sup class="m-sup">4</sup><span class="c-z">z</span><sup class="m-sup">&minus;2</sup></span>
              </span>
            </span>
            <span class="m-den"><span class="c-x">x</span><sup class="m-sup">2</sup><span class="c-y">y</span><sup class="m-sup">1/2</sup></span>
          </span>
          <span class="m-eq">&rarr;</span>
          <span class="m-frac">
            <span class="m-num">(<span class="c-coef">27</span><span class="c-x">x</span><sup class="m-sup">6</sup><span class="c-y">y</span><sup class="m-sup">4</sup><span class="c-z">z</span><sup class="m-sup">&minus;2</sup>)<sup class="m-sup">1/3</sup></span>
            <span class="m-den"><span class="c-x">x</span><sup class="m-sup">2</sup><span class="c-y">y</span><sup class="m-sup">1/2</sup></span>
          </span>
        </div>
      </div>
      <div class="pipeline-connector">
        <div class="connector-badge">&darr;</div>
      </div>
    </div>

    
    <div class="card-wrapper">
      <div class="step-card" data-step="1">
        <div class="card-header">
          <span class="badge">Step 2</span>
          <span class="flow-tag">Multiply outer <sup>1/3</sup> power</span>
        </div>
        <div class="card-title">Distribute power to all factors:</div>
        <div class="math-content">
          <span class="m-frac">
            <span class="m-num">
              <span class="c-coef">27</span><sup class="m-sup">1/3</sup> &middot; 
              <span class="c-x">x</span><sup class="m-sup">6 &middot; 1/3</sup> &middot; 
              <span class="c-y">y</span><sup class="m-sup">4 &middot; 1/3</sup> &middot; 
              <span class="c-z">z</span><sup class="m-sup">&minus;2 &middot; 1/3</sup>
            </span>
            <span class="m-den"><span class="c-x">x</span><sup class="m-sup">2</sup><span class="c-y">y</span><sup class="m-sup">1/2</sup></span>
          </span>
          <span class="m-eq">=</span>
          <span class="m-frac">
            <span class="m-num"><span class="c-coef">3</span><span class="c-x">x</span><sup class="m-sup">2</sup><span class="c-y">y</span><sup class="m-sup">4/3</sup><span class="c-z">z</span><sup class="m-sup">&minus;2/3</sup></span>
            <span class="m-den"><span class="c-x">x</span><sup class="m-sup">2</sup><span class="c-y">y</span><sup class="m-sup">1/2</sup></span>
          </span>
        </div>
      </div>
      <div class="pipeline-connector">
        <div class="connector-badge">&darr;</div>
      </div>
    </div>

    
    <div class="card-wrapper">
      <div class="step-card" data-step="2">
        <div class="card-header">
          <span class="badge">Step 3</span>
          <span class="flow-tag">Subtract matching powers</span>
        </div>
        <div class="card-title">Group matching variable bases:</div>
        <div class="math-content">
          <span class="c-coef">3</span> &middot; 
          <span class="c-x">x</span><sup class="m-sup">2 &minus; 2</sup> &middot; 
          <span class="c-y">y</span><sup class="m-sup"><span class="m-frac"><span class="m-num">4</span><span class="m-den">3</span></span> &minus; <span class="m-frac"><span class="m-num">1</span><span class="m-den">2</span></span></sup> &middot; 
          <span class="c-z">z</span><sup class="m-sup">&minus;2/3</sup>
          <span class="m-eq">=</span>
          <span class="c-coef">3</span> &middot; 
          <span class="c-x">x</span><sup class="m-sup">0</sup> &middot; 
          <span class="c-y">y</span><sup class="m-sup">5/6</sup> &middot; 
          <span class="c-z">z</span><sup class="m-sup">&minus;2/3</sup>
        </div>
      </div>
      <div class="pipeline-connector">
        <div class="connector-badge">&darr;</div>
      </div>
    </div>

    
    <div class="card-wrapper">
      <div class="step-card" data-step="3">
        <div class="card-header">
          <span class="badge">Step 4</span>
          <span class="flow-tag">&check; Positive exponents</span>
        </div>
        <div class="card-title">Final simplified fraction:</div>
        <div class="math-content">
          <span class="m-frac" style="font-size: 1.15em;">
            <span class="m-num"><span class="c-coef">3</span><span class="c-y">y</span><sup class="m-sup">5/6</sup></span>
            <span class="m-den"><span class="c-z">z</span><sup class="m-sup">2/3</sup></span>
          </span>
          <span style="font-size: 0.85em; color: #059669; font-family: system-ui; font-weight: 700; margin-left: 12px;">(x<sup>0</sup> = 1)</span>
        </div>
      </div>
    </div>

  </div>

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</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Simplifying Rational Expressions with Exponents</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
      -webkit-tap-highlight-color: transparent;
    }

    body {
      background: transparent;
      color: #1e293b;
      padding: 10px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      max-width: 680px;
      margin: 0 auto;
    }

    /* Math formatting helpers */
    .math {
      font-family: 'Cambria Math', 'Times New Roman', Cambria, Georgia, serif;
      font-style: italic;
      font-size: 1.05rem;
      letter-spacing: 0.5px;
      display: inline-flex;
      align-items: center;
      justify-content: center;
      flex-wrap: wrap;
      gap: 3px;
      color: #0f172a;
    }

    .math sup, .math sub {
      font-size: 0.72em;
      line-height: 0;
      position: relative;
      font-style: normal;
    }

    .math sup {
      top: -0.55em;
      margin-left: 1px;
    }

    .math sub {
      bottom: -0.25em;
      margin-left: 1px;
    }

    .math .sym {
      font-style: normal;
      margin: 0 2px;
      color: #64748b;
    }

    .math .num-val {
      font-style: normal;
      font-weight: 600;
    }

    .math .op {
      font-style: normal;
      margin: 0 4px;
      font-weight: 500;
      color: #64748b;
    }

    .frac {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      padding: 0 2px;
      font-size: 0.9em;
      font-style: normal;
    }

    .frac-top {
      border-bottom: 1.5px solid currentColor;
      padding: 0 2px 1px 2px;
      line-height: 1.1;
    }

    .frac-bottom {
      padding: 1px 2px 0 2px;
      line-height: 1.1;
    }

    .sqrt {
      display: inline-flex;
      align-items: stretch;
      vertical-align: middle;
      font-style: normal;
    }

    .sqrt-radical {
      font-size: 1.2em;
      line-height: 1;
      padding-right: 1px;
    }

    .sqrt-content {
      border-top: 1.5px solid currentColor;
      padding: 2px 3px 0 2px;
    }

    /* Stepper Navigation */
    .nav-bar {
      display: flex;
      justify-content: center;
      gap: 6px;
      flex-wrap: wrap;
    }

    .step-btn {
      background: #ffffff;
      border: 1.5px solid #cbd5e1;
      padding: 6px 12px;
      border-radius: 20px;
      font-size: 12px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 5px;
      transition: border-color 0.2s, color 0.2s, background-color 0.2s, box-shadow 0.2s;
    }

    .step-btn:hover {
      background-color: #f8fafc;
      border-color: #94a3b8;
    }

    .step-btn.active {
      background: #f0f9ff;
      border-color: #22b7ff;
      color: #0284c7;
      box-shadow: 0 0 0 1px #22b7ff;
    }

    /* Side-by-Side Grid */
    .split-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 10px;
    }

    @media (max-width: 480px) {
      .split-grid {
        grid-template-columns: 1fr;
      }
    }

    /* Card Panels */
    .panel {
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px 12px;
      display: flex;
      flex-direction: column;
      gap: 8px;
      transition: border-color 0.25s, box-shadow 0.25s;
    }

    .panel.highlight {
      border-color: #22b7ff;
      box-shadow: 0 2px 10px rgba(34, 183, 255, 0.18);
    }

    .panel-header {
      display: flex;
      align-items: center;
      justify-content: space-between;
      border-bottom: 1px solid #f1f5f9;
      padding-bottom: 6px;
    }

    .badge {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      padding: 2px 7px;
      border-radius: 6px;
    }

    .badge.num { background: #e0f2fe; color: #0369a1; }
    .badge.den { background: #fef3c7; color: #b45309; }
    .badge.res { background: #ecfdf5; color: #047857; }

    .header-desc {
      font-size: 11px;
      color: #64748b;
      font-weight: 600;
    }

    .math-row {
      display: flex;
      align-items: center;
      justify-content: center;
      min-height: 48px;
      padding: 6px 8px;
      border-radius: 6px;
      background: #f8f9fa;
      text-align: center;
    }

    .step-explain {
      font-size: 11px;
      color: #64748b;
      line-height: 1.35;
      text-align: center;
    }

    /* Bottom Panel */
    .bottom-panel {
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px 12px;
      display: flex;
      flex-direction: column;
      gap: 8px;
      transition: border-color 0.25s, box-shadow 0.25s;
    }

    .bottom-panel.highlight {
      border-color: #22b7ff;
      box-shadow: 0 2px 12px rgba(34, 183, 255, 0.2);
    }

    .bottom-content {
      background: #f8f9fa;
      border-radius: 6px;
      padding: 8px 10px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      gap: 6px;
      min-height: 52px;
    }

    .final-chip {
      background: #f0fdf4;
      border: 1.5px solid #22c55e;
      border-radius: 6px;
      padding: 3px 8px;
      display: inline-flex;
      align-items: center;
    }

    .final-chip-blue {
      background: #e0f7ff;
      border: 1.5px solid #22b7ff;
      border-radius: 6px;
      padding: 3px 8px;
      display: inline-flex;
      align-items: center;
    }

    .note-text {
      font-size: 11px;
      color: #64748b;
      text-align: center;
    }
  </style>
</head>
<body>

  
  <div class="nav-bar">
    <button class="step-btn active" id="btn-0" onclick="setStep(0)">1. Distribute Powers</button>
    <button class="step-btn" id="btn-1" onclick="setStep(1)">2. Simplify Terms</button>
    <button class="step-btn" id="btn-2" onclick="setStep(2)">3. Divide &amp; Combine</button>
  </div>

  
  <div class="split-grid">
    
    <div class="panel" id="num-panel">
      <div class="panel-header">
        <span class="badge num">Numerator</span>
        <span class="header-desc" id="num-sub">Step 1 of 3</span>
      </div>
      <div class="math-row">
        <div class="math" id="num-math"></div>
      </div>
      <div class="step-explain" id="num-note">Apply power 3/4 to each factor</div>
    </div>

    
    <div class="panel" id="den-panel">
      <div class="panel-header">
        <span class="badge den">Denominator</span>
        <span class="header-desc" id="den-sub">Step 1 of 3</span>
      </div>
      <div class="math-row">
        <div class="math" id="den-math"></div>
      </div>
      <div class="step-explain" id="den-note">Rewrite square root as exponent 1/2</div>
    </div>
  </div>

  
  <div class="bottom-panel" id="div-panel">
    <div class="panel-header">
      <span class="badge res">Combined Fraction</span>
      <span class="header-desc" id="div-sub">Subtract powers: <i>a<sup>m</sup> / a<sup>n</sup> = a<sup>m−n</sup></i></span>
    </div>
    <div class="bottom-content">
      <div class="math" id="div-math"></div>
      <div class="note-text" id="div-note">Divide coefficients and subtract corresponding exponents</div>
    </div>
  </div>

  <script>
    const steps = [
      {
        numSub: "Power of a Product",
        numMath: `
          <span class="num-val">16</span><sup><span class="frac"><span class="frac-top">3</span><span class="frac-bottom">4</span></span></sup>
          <span class="sym">·</span>
          <i>x</i><sup>4<span class="sym">·</span><span class="frac"><span class="frac-top">3</span><span class="frac-bottom">4</span></span></sup>
          <span class="sym">·</span>
          <i>y</i><sup>-8<span class="sym">·</span><span class="frac"><span class="frac-top">3</span><span class="frac-bottom">4</span></span></sup>
        `,
        numNote: "Distribute exponent 3/4 to 16, x⁴, and y⁻⁸",
        denSub: "Rewrite Radical as Power",
        denMath: `
          <span class="num-val">4</span><sup><span class="frac"><span class="frac-top">1</span><span class="frac-bottom">2</span></span></sup>
          <span class="sym">·</span>
          <i>x</i><sup>-2<span class="sym">·</span><span class="frac"><span class="frac-top">1</span><span class="frac-bottom">2</span></span></sup>
          <span class="sym">·</span>
          <i>y</i><sup>2<span class="sym">·</span><span class="frac"><span class="frac-top">1</span><span class="frac-bottom">2</span></span></sup>
        `,
        denNote: "Rewrite √(4x⁻²y²) as (4x⁻²y²)¹ᐟ² and distribute 1/2",
        divMath: `
          <span class="frac">
            <span class="frac-top">(16<i>x</i><sup>4</sup><i>y</i><sup>-8</sup>)<sup><span class="frac"><span class="frac-top">3</span><span class="frac-bottom">4</span></span></sup></span>
            <span class="frac-bottom"><span class="sqrt"><span class="sqrt-radical">√</span><span class="sqrt-content">4<i>x</i><sup>-2</sup><i>y</i><sup>2</sup></span></span></span>
          </span>
        `,
        divNote: "Original rational expression before simplification",
        active: "sides"
      },
      {
        numSub: "Simplified Numerator",
        numMath: `
          <span class="final-chip-blue">
            <span class="num-val" style="color:#0369a1;">8</span>
            <i>x</i><sup>3</sup>
            <i>y</i><sup>-6</sup>
          </span>
        `,
        numNote: "16³ᐟ⁴ = 8, 4·(3/4) = 3, -8·(3/4) = -6",
        denSub: "Simplified Denominator",
        denMath: `
          <span class="final-chip-blue">
            <span class="num-val" style="color:#0369a1;">2</span>
            <i>x</i><sup>-1</sup>
            <i>y</i><sup>1</sup>
          </span>
        `,
        denNote: "√4 = 2, -2·(1/2) = -1, 2·(1/2) = 1",
        divMath: `
          <span class="frac">
            <span class="frac-top">8<i>x</i><sup>3</sup><i>y</i><sup>-6</sup></span>
            <span class="frac-bottom">2<i>x</i><sup>-1</sup><i>y</i><sup>1</sup></span>
          </span>
        `,
        divNote: "Substitute simplified numerator and denominator back into quotient",
        active: "sides"
      },
      {
        numSub: "8x³y⁻⁶",
        numMath: `<span class="num-val">8</span><i>x</i><sup>3</sup><i>y</i><sup>-6</sup>`,
        numNote: "Numerator ready for quotient division",
        denSub: "2x⁻¹y¹",
        denMath: `<span class="num-val">2</span><i>x</i><sup>-1</sup><i>y</i>`,
        denNote: "Denominator ready for quotient division",
        divMath: `
          <span class="frac"><span class="frac-top">8</span><span class="frac-bottom">2</span></span>
          <span class="sym">·</span>
          <i>x</i><sup>3 <span class="op">-</span> (<span class="op">-</span>1)</sup>
          <span class="sym">·</span>
          <i>y</i><sup>-6 <span class="op">-</span> 1</sup>
          <span class="op">=</span>
          <span class="num-val">4</span><i>x</i><sup>4</sup><i>y</i><sup>-7</sup>
          <span class="op">=</span>
          <span class="final-chip">
            <span class="frac">
              <span class="frac-top" style="color:#15803d; font-weight:600;">4<i>x</i><sup>4</sup></span>
              <span class="frac-bottom" style="color:#15803d; font-weight:600;"><i>y</i><sup>7</sup></span>
            </span>
          </span>
        `,
        divNote: "Final result written with positive exponents only",
        active: "bottom"
      }
    ];

    let currentStep = 0;

    function render() {
      const data = steps[currentStep];

      // Update button states
      for (let i = 0; i < 3; i++) {
        const btn = document.getElementById(`btn-${i}`);
        if (btn) {
          btn.classList.toggle('active', i === currentStep);
        }
      }

      // Update contents
      document.getElementById('num-sub').textContent = data.numSub;
      document.getElementById('num-math').innerHTML = data.numMath;
      document.getElementById('num-note').textContent = data.numNote;

      document.getElementById('den-sub').textContent = data.denSub;
      document.getElementById('den-math').innerHTML = data.denMath;
      document.getElementById('den-note').textContent = data.denNote;

      document.getElementById('div-math').innerHTML = data.divMath;
      document.getElementById('div-note').textContent = data.divNote;

      // Update highlights
      const numPanel = document.getElementById('num-panel');
      const denPanel = document.getElementById('den-panel');
      const divPanel = document.getElementById('div-panel');

      if (data.active === 'sides') {
        numPanel.classList.add('highlight');
        denPanel.classList.add('highlight');
        divPanel.classList.remove('highlight');
      } else {
        numPanel.classList.remove('highlight');
        denPanel.classList.remove('highlight');
        divPanel.classList.add('highlight');
      }
    }

    function setStep(index) {
      currentStep = index;
      render();
    }

    // Initial render
    render();
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Rationalizing Denominators</title>
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <script src="{{VENDOR_CDN}}/classace-diagram.js"></script>
  <style>
    :root {
      --primary: #22b7ff;
      --primary-dark: #0284c7;
      --primary-light: #f0f9ff;
      --primary-tint: #e0f2fe;
      --text: #0f172a;
      --text-muted: #64748b;
      --border: #e6e6e6;
      --border-dark: #cbd5e1;
      --bg-card: #ffffff;
      --step-inactive-bg: #f8fafc;
      --step-inactive-border: #e2e8f0;
      --success: #059669;
    }
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
      color: var(--text);
      background: transparent;
      padding: 6px;
      user-select: none;
      -webkit-user-select: none;
    }
    .container {
      width: 100%;
      max-width: 720px;
      margin: 0 auto;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }
    .main-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 10px;
      width: 100%;
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    @media (max-width: 540px) {
      .main-grid {
        grid-template-columns: 1fr;
        gap: 8px;
      }
    }
    .math-card {
      background: var(--bg-card);
      border: 1.5px solid var(--border);
      border-radius: 8px;
      padding: 8px;
      display: flex;
      flex-direction: column;
      gap: 6px;
    }
    .card-header {
      display: flex;
      align-items: center;
      justify-content: space-between;
      border-bottom: 1px solid var(--border);
      padding-bottom: 5px;
    }
    .card-title {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      color: var(--text);
    }
    .card-tag {
      font-size: 9.5px;
      font-weight: 700;
      background: var(--primary-tint);
      color: var(--primary-dark);
      padding: 2px 6px;
      border-radius: 4px;
      white-space: nowrap;
    }
    .steps-container {
      display: flex;
      flex-direction: column;
      gap: 6px;
    }
    .step-box {
      width: 100%;
      padding: 6px 8px;
      border-radius: 6px;
      background: var(--step-inactive-bg);
      border: 1.5px solid var(--step-inactive-border);
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      text-align: center;
      transition: background-color 0.2s ease, border-color 0.2s ease, box-shadow 0.2s ease, opacity 0.2s ease;
      opacity: 0.75;
      cursor: pointer;
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    .step-box:hover {
      opacity: 0.95;
      border-color: var(--border-dark);
    }
    .step-box.active {
      opacity: 1;
      background: var(--primary-light);
      border-color: var(--primary);
      box-shadow: 0 0 0 1px var(--primary);
    }
    .step-box.done {
      opacity: 1;
      background: #ffffff;
      border-color: var(--border-dark);
    }
    .step-header {
      display: flex;
      align-items: center;
      gap: 5px;
      margin-bottom: 3px;
      width: 100%;
      justify-content: center;
    }
    .step-num {
      display: inline-flex;
      align-items: center;
      justify-content: center;
      width: 15px;
      height: 15px;
      font-size: 9px;
      font-weight: 700;
      border-radius: 50%;
      background: #cbd5e1;
      color: #1e293b;
      flex-shrink: 0;
    }
    .active .step-num {
      background: var(--primary-dark);
      color: #ffffff;
    }
    .done .step-num {
      background: var(--text-muted);
      color: #ffffff;
    }
    .step-label {
      font-size: 10.5px;
      font-weight: 600;
      color: var(--text-muted);
      white-space: nowrap;
      overflow: hidden;
      text-overflow: ellipsis;
    }
    .active .step-label {
      color: var(--primary-dark);
      font-weight: 700;
    }
    .math-content {
      font-size: 13px;
      color: var(--text);
      line-height: 1.2;
      width: 100%;
      display: flex;
      justify-content: center;
      align-items: center;
      overflow-x: auto;
      overflow-y: hidden;
      padding: 2px 0;
    }
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      font-size: 0.95em;
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  </style>
</head>
<body>

  <div class="container">
    <div class="main-grid">
      
      <div class="math-card">
        <div class="card-header">
          <span class="card-title">Monomial Denominator</span>
          <span class="card-tag">Power Rule</span>
        </div>
        
        <div class="steps-container">
          
          <div class="step-box step-item" data-step="0" onclick="goToStep(0)">
            <div class="step-header">
              <span class="step-num">1</span>
              <span class="step-label">Initial Expression</span>
            </div>
            <div class="math-content">
              <div class="katex-eq" data-tex="\displaystyle \frac{1}{x^{2/5}}"></div>
            </div>
          </div>

          
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            <div class="step-header">
              <span class="step-num">2</span>
              <span class="step-label">Multiply top &amp; bottom by x^(3/5)</span>
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            <div class="math-content">
              <div class="katex-eq" data-tex="\displaystyle \frac{1}{x^{2/5}} \cdot {\color{#0284c7}{\frac{x^{3/5}}{x^{3/5}}}}"></div>
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            <div class="sub-note">Power sum: 2/5 + 3/5 = 1</div>
          </div>

          
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            <div class="step-header">
              <span class="step-num">3</span>
              <span class="step-label">Rationalized Result</span>
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            <div class="math-content">
              <div class="katex-eq" data-tex="\displaystyle \frac{x^{3/5}}{x^{2/5 + 3/5}} = {\mathbf{\frac{x^{3/5}}{x}}}"></div>
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      </div>

      
      <div class="math-card">
        <div class="card-header">
          <span class="card-title">Binomial Denominator</span>
          <span class="card-tag">Conjugate Pair</span>
        </div>

        <div class="steps-container">
          
          <div class="step-box step-item" data-step="0" onclick="goToStep(0)">
            <div class="step-header">
              <span class="step-num">1</span>
              <span class="step-label">Initial Expression</span>
            </div>
            <div class="math-content">
              <div class="katex-eq" data-tex="\displaystyle \frac{1}{3 + \sqrt{2}}"></div>
            </div>
          </div>

          
          <div class="step-box step-item" data-step="1" onclick="goToStep(1)">
            <div class="step-header">
              <span class="step-num">2</span>
              <span class="step-label">Multiply by conjugate (3 - √2)</span>
            </div>
            <div class="math-content">
              <div class="katex-eq" data-tex="\displaystyle \frac{1}{3 + \sqrt{2}} \cdot {\color{#0284c7}{\frac{3 - \sqrt{2}}{3 - \sqrt{2}}}}"></div>
            </div>
            <div class="sub-note">Conjugate of (3 + √2) is (3 - √2)</div>
          </div>

          
          <div class="step-box step-item" data-step="2" onclick="goToStep(2)">
            <div class="step-header">
              <span class="step-num">3</span>
              <span class="step-label">Difference of Squares</span>
            </div>
            <div class="math-content">
              <div class="katex-eq" data-tex="\displaystyle \frac{3 - \sqrt{2}}{3^2 - (\sqrt{2})^2} = \frac{3 - \sqrt{2}}{9 - 2} = {\mathbf{\frac{3 - \sqrt{2}}{7}}}"></div>
            </div>
            <div class="sub-note">3² - (√2)² = 9 - 2 = 7</div>
          </div>
        </div>
      </div>
    </div>

    
    <div class="controls-bar">
      <button class="nav-btn" id="prevBtn" title="Previous step">◀ Prev</button>
      <button class="nav-btn" id="playBtn"><span id="playIcon">⏸</span> Pause</button>
      <button class="nav-btn" id="nextBtn" title="Next step">Next ▶</button>
      <div class="dots-container" aria-label="Step indicator">
        <div class="dot active" data-idx="0" title="Step 1: Initial Expression"></div>
        <div class="dot" data-idx="1" title="Step 2: Multiply"></div>
        <div class="dot" data-idx="2" title="Step 3: Simplify"></div>
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<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Radical to Fractional Exponent Transformation</title>
  <style>
    * { 
      box-sizing: border-box; 
      margin: 0; 
      padding: 0; 
    }
    
    body {
      width: 100%;
      background: #ffffff;
      color: #1e2945;
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    sup {
      font-style: normal;
      font-size: 0.62em;
      vertical-align: 0.55em;
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    .frac {
      display: inline-flex;
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      text-align: center;
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    .frac .num {
      border-bottom: 1.5px solid currentColor;
      padding-bottom: 1px;
    }
    
    .frac .denom {
      padding-top: 1px;
    }

    /* Reference Rule Banner */
    .rule-banner {
      display: flex;
      align-items: center;
      justify-content: center;
      flex-wrap: wrap;
      gap: 8px 14px;
      padding: 8px 16px;
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      border: 1px solid #cbd5e1;
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      font-weight: 600;
      max-width: 100%;
    }

    .rule-formula {
      display: inline-flex;
      align-items: center;
      gap: 6px;
      font-size: 16px;
      font-weight: bold;
    }

    .banner-radical {
      display: inline-flex;
      align-items: center;
      position: relative;
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    .banner-index {
      font-size: 11px;
      color: #0284c7;
      font-weight: 800;
      margin-right: 2px;
      margin-top: -6px;
    }

    .banner-radicand {
      border-top: 2px solid #1e2945;
      padding: 1px 4px 0 3px;
      margin-left: -2px;
      line-height: 1.2;
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    /* Badges */
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      display: inline-flex;
      align-items: center;
      gap: 6px;
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      display: inline-flex;
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      font-size: 11px;
      font-weight: 700;
      letter-spacing: 0.02em;
      white-space: nowrap;
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      background: #e0f2fe; 
      color: #0369a1; 
      border: 1px solid #bae6fd;
    }
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      color: #92400e; 
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      gap: 6px;
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      gap: 8px;
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      .radical-index { font-size: 11px; margin-top: -8px; }
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  </style>
</head>
<body>

  
  <div class="rule-banner">
    <div class="rule-formula math-font">
      <span class="banner-radical">
        <span class="banner-index color-root">root</span>
        <svg class="radical-svg" viewBox="0 0 16 36" preserveAspectRatio="none" style="height: 24px; width: 11px;">
          <path d="M1,20 L4,20 L7,34 L15,2" fill="none" stroke="#1e2945" stroke-width="2.2" stroke-linecap="round" stroke-linejoin="round"/>
        </svg>
        <span class="banner-radicand">a<sup class="color-power">power</sup></span>
      </span>
      <span>=</span>
      <span>a<sup class="frac"><span class="num color-power">power</span><span class="denom color-root">root</span></sup></span>
    </div>
    <div class="badge-group">
      <span class="badge badge-power">Power (Numerator)</span>
      <span class="badge badge-root">Root (Denominator)</span>
    </div>
  </div>

  
  <div class="viewer">
    
    <div class="stage active" data-step="0">
      <div class="step-title">Step 1: Radical Expression</div>
      <div class="math-display math-font">
        <span class="radical-wrap">
          <span class="radical-index">3</span>
          <svg class="radical-svg" viewBox="0 0 16 36" preserveAspectRatio="none">
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          <span class="radicand">8x<sup>6</sup>y<sup>9</sup></span>
        </span>
      </div>
      <div class="breakdown">
        <span class="badge badge-root">Root / Index = 3</span>
        <span class="badge badge-power">Radicand = 8 &middot; x<sup>6</sup> &middot; y<sup>9</sup></span>
      </div>
    </div>

    
    <div class="stage" data-step="1">
      <div class="step-title">Step 2: Convert to Fractional Exponent</div>
      <div class="math-display math-font">
        <span>(8x<sup>6</sup>y<sup>9</sup>)</span><sup class="frac"><span class="num color-power">1</span><span class="denom color-root">3</span></sup>
      </div>
      <div class="breakdown">
        <span class="badge badge-power">Power = 1</span>
        <span class="badge badge-root">Root = 3</span>
        <span class="badge badge-root">Overall Exponent = <sup>1</sup>&frasl;<sub>3</sub></span>
      </div>
    </div>

    
    <div class="stage" data-step="2">
      <div class="step-title">Step 3: Distribute Exponent & Simplify</div>
      <div class="math-display math-font">
        8<sup class="frac"><span class="num">1</span><span class="denom">3</span></sup> &middot; x<sup class="frac"><span class="num color-power">6</span><span class="denom color-root">3</span></sup> &middot; y<sup class="frac"><span class="num color-power">9</span><span class="denom color-root">3</span></sup> = <span class="color-result">2x<sup>2</sup>y<sup>3</sup></span>
      </div>
      <div class="breakdown">
        <span class="badge badge-result">8<sup>1/3</sup> = 2</span>
        <span class="badge badge-result">x<sup>6/3</sup> = x<sup>2</sup></span>
        <span class="badge badge-result">y<sup>9/3</sup> = y<sup>3</sup></span>
      </div>
    </div>
  </div>

  
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    <button class="step-btn active" data-index="0">1 Radical</button>
    <button class="step-btn" data-index="1">2 Exponent</button>
    <button class="step-btn" data-index="2">3 Simplified</button>
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</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
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      color: #0f172a;
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</head>
<body>

  
  <div class="header">
    <h1>The Equivalence Principle: Rational Exponent to Radical</h1>
    <p>The denominator is the root index &bull; The numerator is the power</p>
  </div>

  
  <div class="anatomy-section">
    <div class="pill-row">
      <span class="badge-pill badge-power">Numerator <strong class="coral-text">m</strong> → Power / Exponent</span>
      <span class="badge-pill badge-root">Denominator <strong class="teal-text">n</strong> → Root Index</span>
    </div>
    
    
    <div class="formula-display">
      
      <span class="math-term">
        <span>x</span><span class="exp-frac"><span class="num coral-text">m</span><span class="den teal-text">n</span></span>
      </span>

      <span class="math-eq">=</span>

      
      <span class="math-term">
        <span class="radical-block">
          <span class="radical-index teal-text">n</span>
          <span class="radical-symbol">√</span>
          <span class="radical-radicand">x<sup class="sup-pow coral-text">m</sup></span>
        </span>
      </span>

      <span class="math-eq">=</span>

      
      <span class="math-term">
        <span>(</span><span class="radical-block"><span class="radical-index teal-text">n</span><span class="radical-symbol">√</span><span class="radical-radicand">x</span></span><span>)</span><sup class="sup-pow coral-text" style="top:-8px;">m</sup>
      </span>
    </div>
  </div>

  
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    <button class="tab-btn" id="tabB" onclick="setMode('B')">Option B: Power First</button>
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  <div class="pathways-wrapper" id="pathways">
    
    
    <div class="path-card highlight" id="cardA">
      <div class="path-header">
        <span class="path-title">Pathway A: Root First</span>
        <span class="path-tag tag-recommended">✦ Recommended for Mental Math</span>
      </div>
      <div class="steps-row">
        
        <div class="step-box">
          <span class="step-label">Step 1: Setup</span>
          <span class="step-math">
            <span>8</span><span class="exp-frac"><span class="num coral-text">2</span><span class="den teal-text">3</span></span>
            <span style="font-style:normal; margin:0 4px; font-size:12px; color:#64748b;">=</span>
            <span>(</span><span class="radical-block"><span class="radical-index teal-text">3</span><span class="radical-symbol">√</span><span class="radical-radicand">8</span></span><span>)</span><sup class="sup-pow coral-text">2</sup>
          </span>
        </div>
        <span class="step-arrow">→</span>
        
        <div class="step-box">
          <span class="step-label">Step 2: Root</span>
          <span class="step-math">
            <span>(<strong>2</strong>)</span><sup class="sup-pow coral-text">2</sup>
          </span>
        </div>
        <span class="step-arrow">→</span>
        
        <div class="step-box">
          <span class="step-label">Step 3: Result</span>
          <span class="step-math">
            <span class="result-pill">4</span>
          </span>
        </div>
      </div>
      <div class="note-footer">
        <span>💡</span>
        <span>Numbers stay small: <strong>∛8 = 2</strong>, then evaluate <strong>2² = 4</strong></span>
      </div>
    </div>

    
    <div class="path-card" id="cardB" style="display: none;">
      <div class="path-header">
        <span class="path-title">Pathway B: Power First</span>
        <span class="path-tag tag-alternative">Alternative (Larger Numbers)</span>
      </div>
      <div class="steps-row">
        
        <div class="step-box">
          <span class="step-label">Step 1: Setup</span>
          <span class="step-math">
            <span>8</span><span class="exp-frac"><span class="num coral-text">2</span><span class="den teal-text">3</span></span>
            <span style="font-style:normal; margin:0 4px; font-size:12px; color:#64748b;">=</span>
            <span class="radical-block"><span class="radical-index teal-text">3</span><span class="radical-symbol">√</span><span class="radical-radicand">8<sup class="sup-pow coral-text">2</sup></span></span>
          </span>
        </div>
        <span class="step-arrow">→</span>
        
        <div class="step-box">
          <span class="step-label">Step 2: Power</span>
          <span class="step-math">
            <span class="radical-block"><span class="radical-index teal-text">3</span><span class="radical-symbol">√</span><span class="radical-radicand"><strong>64</strong></span></span>
          </span>
        </div>
        <span class="step-arrow">→</span>
        
        <div class="step-box">
          <span class="step-label">Step 3: Result</span>
          <span class="step-math">
            <span class="result-pill">4</span>
          </span>
        </div>
      </div>
      <div class="note-footer">
        <span>⚠️</span>
        <span>Numbers grow large: <strong>8² = 64</strong>, then evaluate <strong>∛64 = 4</strong></span>
      </div>
    </div>

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OGM.13.NUM.2

Aligned learning resources

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