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Tutoring abilities

Imagine writing $2 \times 2 \times 2 \times 2 \times 2 \times 2$. Your hand would get tired very fast!

Math has a clever shortcut for multiplying the same number again and again called **repeated multiplication**.

Instead of writing a long string of multiplication, we pack it into a short code called a **power**.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Understanding Powers</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      user-select: none;
      -webkit-user-select: none;
    }

    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100vh;
      padding: 16px 12px;
      overflow-x: hidden;
    }

    .controls-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      width: 100%;
      max-width: 480px;
      flex-wrap: wrap;
    }

    .pill-group {
      display: flex;
      background: #e9ecef;
      padding: 3px;
      border-radius: 20px;
      gap: 2px;
    }

    .pill-btn {
      border: none;
      background: transparent;
      padding: 6px 14px;
      border-radius: 16px;
      font-size: 14px;
      font-weight: 600;
      color: #6c757d;
      cursor: pointer;
      transition: background 0.2s, color 0.2s, box-shadow 0.2s;
    }

    .pill-btn.active {
      background: #ffffff;
      color: #0284c7;
      box-shadow: 0 2px 6px rgba(0,0,0,0.08);
    }

    .stepper {
      display: flex;
      align-items: center;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 20px;
      padding: 2px 6px;
      gap: 8px;
    }

    .step-btn {
      width: 28px;
      height: 28px;
      border-radius: 50%;
      border: none;
      background: #f1f5f9;
      color: #0f172a;
      font-size: 16px;
      font-weight: bold;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
      transition: background 0.15s;
    }

    .step-btn:disabled {
      opacity: 0.3;
      cursor: not-allowed;
    }

    .step-btn:not(:disabled):hover {
      background: #e2e8f0;
    }

    .stage {
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      width: 100%;
      max-width: 520px;
      flex: 1;
      gap: 18px;
      margin: 12px 0;
    }

    .string-container {
      display: flex;
      align-items: center;
      justify-content: center;
      flex-wrap: wrap;
      gap: 6px;
      background: #ffffff;
      padding: 16px 14px;
      border-radius: 16px;
      box-shadow: 0 4px 12px rgba(0,0,0,0.04);
      min-height: 84px;
      width: 100%;
    }

    .factor-item {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 4px;
      animation: fadeIn 0.3s ease;
    }

    @keyframes fadeIn {
      from { opacity: 0; transform: translateY(6px); }
      to { opacity: 1; transform: translateY(0); }
    }

    .factor-box {
      width: 38px;
      height: 38px;
      border-radius: 10px;
      background: #e0f2fe;
      border: 2px solid #38bdf8;
      color: #0369a1;
      font-size: 18px;
      font-weight: 700;
      display: flex;
      align-items: center;
      justify-content: center;
    }

    .count-tag {
      font-size: 11px;
      font-weight: 700;
      color: #f43f5e;
      background: #ffe4e6;
      border-radius: 8px;
      padding: 1px 6px;
    }

    .mult-sign {
      color: #94a3b8;
      font-size: 16px;
      font-weight: 700;
      margin-top: -14px;
    }

    .transform-arrow {
      color: #94a3b8;
      font-size: 22px;
      display: flex;
      align-items: center;
      justify-content: center;
      height: 24px;
    }

    .power-container {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 16px;
      background: #ffffff;
      padding: 18px 24px;
      border-radius: 16px;
      box-shadow: 0 4px 12px rgba(0,0,0,0.04);
      width: 100%;
    }

    .power-code {
      display: inline-flex;
      align-items: flex-start;
      line-height: 1;
    }

    .base-val {
      font-size: 42px;
      font-weight: 800;
      color: #0284c7;
      background: #e0f2fe;
      border-radius: 12px;
      padding: 2px 10px;
      border: 2px solid #38bdf8;
      line-height: 1.1;
    }

    .exp-val {
      font-size: 24px;
      font-weight: 800;
      color: #e11d48;
      background: #ffe4e6;
      border-radius: 8px;
      padding: 2px 8px;
      border: 2px solid #fb7185;
      margin-left: 4px;
      margin-top: -6px;
    }

    .equals {
      font-size: 28px;
      font-weight: 700;
      color: #94a3b8;
    }

    .eval-val {
      font-size: 32px;
      font-weight: 800;
      color: #1e2945;
      min-width: 48px;
    }

    .legend {
      display: flex;
      justify-content: center;
      gap: 16px;
      width: 100%;
      font-size: 12px;
      font-weight: 600;
    }

    .legend-item {
      display: flex;
      align-items: center;
      gap: 6px;
    }

    .legend-dot {
      width: 10px;
      height: 10px;
      border-radius: 3px;
    }

    .dot-base {
      background: #0284c7;
    }

    .dot-exp {
      background: #e11d48;
    }
  </style>
</head>
<body>

  <div class="controls-bar">
    <div class="pill-group" id="baseSelector">
      <button class="pill-btn active" data-base="2">2</button>
      <button class="pill-btn" data-base="3">3</button>
      <button class="pill-btn" data-base="5">5</button>
      <button class="pill-btn" data-base="10">10</button>
    </div>

    <div class="stepper">
      <button class="step-btn" id="btnMinus" aria-label="Decrease">−</button>
      <span style="font-size: 13px; font-weight: 700; color: #475569; min-width: 50px; text-align: center;" id="repeatText">4 times</span>
      <button class="step-btn" id="btnPlus" aria-label="Increase">+</button>
    </div>
  </div>

  <div class="stage">
    <div class="string-container" id="factorsRow"></div>

    <div class="transform-arrow">↓</div>

    <div class="power-container">
      <div class="power-code">
        <span class="base-val" id="baseDisplay">2</span>
        <span class="exp-val" id="expDisplay">4</span>
      </div>
      <div class="equals">=</div>
      <div class="eval-val" id="evalDisplay">16</div>
    </div>
  </div>

  <div class="legend">
    <div class="legend-item">
      <div class="legend-dot dot-base"></div>
      <span>Base (Factor)</span>
    </div>
    <div class="legend-item">
      <div class="legend-dot dot-exp"></div>
      <span>Exponent (Count)</span>
    </div>
  </div>

  <script>
    let base = 2;
    let exp = 4;

    const factorsRow = document.getElementById('factorsRow');
    const baseDisplay = document.getElementById('baseDisplay');
    const expDisplay = document.getElementById('expDisplay');
    const evalDisplay = document.getElementById('evalDisplay');
    const repeatText = document.getElementById('repeatText');
    const btnMinus = document.getElementById('btnMinus');
    const btnPlus = document.getElementById('btnPlus');
    const baseButtons = document.querySelectorAll('#baseSelector .pill-btn');

    function update() {
      baseDisplay.textContent = base;
      expDisplay.textContent = exp;
      evalDisplay.textContent = Math.pow(base, exp).toLocaleString();
      repeatText.textContent = `${exp} times`;

      btnMinus.disabled = exp <= 2;
      btnPlus.disabled = exp >= 6;

      factorsRow.innerHTML = '';
      for (let i = 1; i <= exp; i++) {
        const item = document.createElement('div');
        item.className = 'factor-item';

        const box = document.createElement('div');
        box.className = 'factor-box';
        box.textContent = base;

        const count = document.createElement('div');
        count.className = 'count-tag';
        count.textContent = '#' + i;

        item.appendChild(box);
        item.appendChild(count);
        factorsRow.appendChild(item);

        if (i < exp) {
          const mult = document.createElement('div');
          mult.className = 'mult-sign';
          mult.textContent = '×';
          factorsRow.appendChild(mult);
        }
      }
    }

    baseButtons.forEach(btn => {
      btn.addEventListener('click', () => {
        baseButtons.forEach(b => b.classList.remove('active'));
        btn.classList.add('active');
        base = parseInt(btn.dataset.base, 10);
        update();
      });
    });

    btnMinus.addEventListener('click', () => {
      if (exp > 2) {
        exp--;
        update();
      }
    });

    btnPlus.addEventListener('click', () => {
      if (exp < 6) {
        exp++;
        update();
      }
    });

    update();
  </script>
</body>
</html>

How do we read the parts of this new math code?

The big number at the bottom is the **base**. This is the number you are multiplying.

The small number floating up top is the **exponent** (also called the **power** or **index**). It tells you how many times to multiply the base.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
    }
    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 16px;
      user-select: none;
    }
    .diagram-content {
      width: 100%;
      max-width: 520px;
      display: flex;
      flex-direction: column;
      gap: 16px;
      align-items: center;
    }

    /* Controls Bar */
    .controls {
      display: flex;
      flex-wrap: wrap;
      align-items: center;
      justify-content: center;
      gap: 12px;
      width: 100%;
    }
    .control-group {
      display: flex;
      align-items: center;
      gap: 6px;
      background: #ffffff;
      padding: 4px 10px;
      border-radius: 20px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.06);
    }
    .control-label {
      font-size: 13px;
      font-weight: 600;
      color: #6b7280;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }
    .btn-chip {
      background: #f1f5f9;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      color: #1e2945;
      font-size: 14px;
      font-weight: 700;
      padding: 4px 10px;
      cursor: pointer;
      transition: background 0.2s, color 0.2s;
    }
    .btn-chip.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
    }
    .btn-step {
      width: 28px;
      height: 28px;
      display: flex;
      align-items: center;
      justify-content: center;
      background: #f1f5f9;
      border: 1px solid #e2e8f0;
      border-radius: 50%;
      font-weight: bold;
      font-size: 15px;
      cursor: pointer;
      color: #1e2945;
      transition: background 0.2s;
    }
    .btn-step:disabled {
      opacity: 0.3;
      cursor: not-allowed;
    }

    /* Power Display Card */
    .power-card {
      background: #ffffff;
      border: 1px solid #e6e6e6;
      border-radius: 16px;
      padding: 20px;
      width: 100%;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
    }

    /* Notated Equation */
    .notation-display {
      display: flex;
      align-items: baseline;
      justify-content: center;
      gap: 12px;
      position: relative;
    }
    .base-box {
      display: inline-flex;
      flex-direction: column;
      align-items: center;
    }
    .base-val {
      font-size: 46px;
      font-weight: 800;
      color: #0284c7;
      line-height: 1;
    }
    .exponent-val {
      font-size: 26px;
      font-weight: 800;
      color: #e11d48;
      background: #ffe4e6;
      border-radius: 6px;
      padding: 2px 8px;
      line-height: 1;
      transform: translateY(-20px);
      margin-left: 2px;
      display: inline-block;
      transition: transform 0.2s ease;
    }

    .tag-label {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      margin-top: 4px;
    }
    .tag-base { color: #0284c7; }
    .tag-exp { color: #e11d48; }

    /* Expansion Visual Row */
    .expansion-container {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 6px;
      width: 100%;
      min-height: 80px;
      justify-content: center;
    }
    .factors-row {
      display: flex;
      align-items: center;
      justify-content: center;
      flex-wrap: wrap;
      gap: 6px;
    }
    .factor-tile {
      background: #e0f2fe;
      border: 2px solid #0284c7;
      color: #0369a1;
      font-size: 22px;
      font-weight: 800;
      width: 40px;
      height: 40px;
      border-radius: 8px;
      display: flex;
      align-items: center;
      justify-content: center;
      animation: popIn 0.3s cubic-bezier(0.175, 0.885, 0.32, 1.275);
    }
    .operator {
      font-size: 20px;
      font-weight: 700;
      color: #94a3b8;
    }

    /* Counter indicator bracket */
    .count-indicator {
      display: flex;
      align-items: center;
      gap: 6px;
      color: #e11d48;
      font-size: 13px;
      font-weight: 700;
      background: #fff1f2;
      border: 1px dashed #f43f5e;
      padding: 3px 12px;
      border-radius: 12px;
    }

    /* Result Row */
    .result-row {
      display: flex;
      align-items: center;
      gap: 10px;
      font-size: 24px;
      font-weight: 800;
      color: #1e2945;
      padding-top: 4px;
      border-top: 1px solid #f1f5f9;
      width: 100%;
      justify-content: center;
    }
    .result-badge {
      background: #ecfdf5;
      color: #059669;
      border: 1.5px solid #10b981;
      padding: 4px 14px;
      border-radius: 10px;
    }

    @keyframes popIn {
      0% { opacity: 0; transform: scale(0.6); }
      100% { opacity: 1; transform: scale(1); }
    }
  </style>
</head>
<body>

  <div class="diagram-content">
    
    <div class="controls">
      <div class="control-group">
        <span class="control-label tag-base">Base</span>
        <button class="btn-chip active" data-base="2">2</button>
        <button class="btn-chip" data-base="3">3</button>
        <button class="btn-chip" data-base="5">5</button>
        <button class="btn-chip" data-base="10">10</button>
      </div>

      <div class="control-group">
        <span class="control-label tag-exp">Power</span>
        <button class="btn-step" id="expDec" aria-label="Decrease power">−</button>
        <span id="expNum" style="font-weight: 800; font-size: 16px; min-width: 14px; text-align: center; color: #e11d48;">3</span>
        <button class="btn-step" id="expInc" aria-label="Increase power">+</button>
      </div>
    </div>

    
    <div class="power-card">
      
      <div class="notation-display">
        <div class="base-box">
          <span class="base-val" id="dispBase">2</span>
          <span class="tag-label tag-base">Base</span>
        </div>
        <div class="base-box" style="margin-left: -4px;">
          <span class="exponent-val" id="dispExp">3</span>
          <span class="tag-label tag-exp">Power</span>
        </div>
      </div>

      
      <div class="expansion-container">
        <div class="factors-row" id="factorsRow"></div>
        <div class="count-indicator" id="countIndicator">
          <span id="countText">3 copies of 2</span>
        </div>
      </div>

      
      <div class="result-row">
        <span>=</span>
        <div class="result-badge" id="resultVal">8</div>
      </div>
    </div>
  </div>

  <script>
    let base = 2;
    let exp = 3;

    const baseChips = document.querySelectorAll('[data-base]');
    const expDecBtn = document.getElementById('expDec');
    const expIncBtn = document.getElementById('expInc');
    const expNum = document.getElementById('expNum');
    const dispBase = document.getElementById('dispBase');
    const dispExp = document.getElementById('dispExp');
    const factorsRow = document.getElementById('factorsRow');
    const countText = document.getElementById('countText');
    const resultVal = document.getElementById('resultVal');

    function render() {
      // Update notation
      dispBase.textContent = base;
      dispExp.textContent = exp;
      expNum.textContent = exp;

      // Update button states
      expDecBtn.disabled = exp <= 1;
      expIncBtn.disabled = exp >= 5;

      // Update factor blocks
      factorsRow.innerHTML = '';
      for (let i = 0; i < exp; i++) {
        const tile = document.createElement('div');
        tile.className = 'factor-tile';
        tile.textContent = base;
        factorsRow.appendChild(tile);

        if (i < exp - 1) {
          const op = document.createElement('div');
          op.className = 'operator';
          op.textContent = '×';
          factorsRow.appendChild(op);
        }
      }

      // Update counter indicator
      countText.textContent = `${exp} × [${base}]`;

      // Update result
      const total = Math.pow(base, exp);
      resultVal.textContent = total.toLocaleString();
    }

    baseChips.forEach(btn => {
      btn.addEventListener('click', () => {
        baseChips.forEach(b => b.classList.remove('active'));
        btn.classList.add('active');
        base = parseInt(btn.getAttribute('data-base'), 10);
        render();
      });
    });

    expDecBtn.addEventListener('click', () => {
      if (exp > 1) {
        exp--;
        render();
      }
    });

    expIncBtn.addEventListener('click', () => {
      if (exp < 5) {
        exp++;
        render();
      }
    });

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

What does this look like when we build one from scratch?

Look at $4 \times 4 \times 4$. The number being multiplied is $4$, so $4$ is our base.

Count how many $4$s there are: one, two, three. That means our exponent is $3$, giving us **power form**: $4^3$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      user-select: none;
      -webkit-user-select: none;
    }

    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 16px;
      gap: 16px;
    }

    /* Controls Bar */
    .controls {
      display: flex;
      flex-wrap: wrap;
      align-items: center;
      justify-content: center;
      gap: 12px;
      width: 100%;
      max-width: 480px;
    }

    .control-group {
      display: flex;
      align-items: center;
      background: #ffffff;
      padding: 4px 6px;
      border-radius: 20px;
      box-shadow: 0 1px 4px rgba(0,0,0,0.06);
      gap: 4px;
    }

    .control-label {
      font-size: 13px;
      font-weight: 600;
      padding: 0 6px;
      color: #6b7280;
    }

    .btn-toggle {
      border: none;
      background: transparent;
      color: #1e2945;
      font-size: 14px;
      font-weight: 700;
      width: 30px;
      height: 30px;
      border-radius: 15px;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      transition: background-color 0.2s ease, color 0.2s ease;
    }

    .btn-toggle.active-base {
      background: #22b7ff;
      color: #ffffff;
    }

    .btn-toggle.active-exp {
      background: #8b5cf6;
      color: #ffffff;
    }

    /* Main Display Card */
    .display-area {
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      width: 100%;
      max-width: 520px;
      gap: 20px;
    }

    /* Power Notation Section */
    .power-view {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 16px;
      height: 90px;
    }

    .power-expression {
      display: inline-flex;
      align-items: flex-start;
      line-height: 1;
      position: relative;
    }

    .base-num {
      font-size: 56px;
      font-weight: 800;
      color: #0284c7;
      position: relative;
    }

    .exp-num {
      font-size: 32px;
      font-weight: 800;
      color: #8b5cf6;
      margin-left: 2px;
      position: relative;
    }

    .tag-label {
      position: absolute;
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      white-space: nowrap;
      padding: 2px 6px;
      border-radius: 4px;
    }

    .tag-base {
      bottom: -20px;
      left: 50%;
      transform: translateX(-50%);
      background: #e0f2fe;
      color: #0369a1;
    }

    .tag-exp {
      top: -18px;
      left: 50%;
      transform: translateX(-50%);
      background: #ede9fe;
      color: #6d28d9;
    }

    .equals-sign {
      font-size: 36px;
      font-weight: 700;
      color: #94a3b8;
    }

    .product-value {
      font-size: 44px;
      font-weight: 800;
      color: #1e2945;
      min-width: 70px;
    }

    /* Factor Expansion Section */
    .expansion-container {
      display: flex;
      flex-direction: column;
      align-items: center;
      width: 100%;
      height: 110px;
      justify-content: center;
    }

    .factors-row {
      display: flex;
      align-items: center;
      justify-content: center;
      flex-wrap: wrap;
      gap: 6px;
    }

    .factor-group {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 4px;
    }

    .count-badge {
      font-size: 11px;
      font-weight: 700;
      color: #8b5cf6;
      background: #ede9fe;
      width: 22px;
      height: 22px;
      border-radius: 50%;
      display: flex;
      align-items: center;
      justify-content: center;
    }

    .factor-tile {
      background: #ffffff;
      border: 2px solid #bae6fd;
      color: #0284c7;
      font-size: 24px;
      font-weight: 800;
      width: 44px;
      height: 48px;
      border-radius: 8px;
      display: flex;
      align-items: center;
      justify-content: center;
      box-shadow: 0 2px 4px rgba(0,0,0,0.04);
      transition: background-color 0.2s ease;
    }

    .multiply-sign {
      font-size: 20px;
      font-weight: 700;
      color: #94a3b8;
      margin-top: 26px;
    }

    .count-summary {
      margin-top: 10px;
      font-size: 13px;
      font-weight: 600;
      color: #6b7280;
    }

    .count-summary span {
      font-weight: 700;
    }
  </style>
</head>
<body>

  
  <div class="controls">
    <div class="control-group">
      <span class="control-label">Base:</span>
      <button class="btn-toggle" data-base="2">2</button>
      <button class="btn-toggle" data-base="3">3</button>
      <button class="btn-toggle active-base" data-base="4">4</button>
      <button class="btn-toggle" data-base="5">5</button>
    </div>

    <div class="control-group">
      <span class="control-label">Exponent:</span>
      <button class="btn-toggle" data-exp="1">1</button>
      <button class="btn-toggle" data-exp="2">2</button>
      <button class="btn-toggle active-exp" data-exp="3">3</button>
      <button class="btn-toggle" data-exp="4">4</button>
      <button class="btn-toggle" data-exp="5">5</button>
    </div>
  </div>

  
  <div class="display-area">
    
    
    <div class="power-view">
      <div class="power-expression">
        <span class="base-num" id="baseDisplay">4
          <span class="tag-label tag-base">Base</span>
        </span>
        <span class="exp-num" id="expDisplay">3
          <span class="tag-label tag-exp">Exponent</span>
        </span>
      </div>
      <span class="equals-sign">=</span>
      <span class="product-value" id="productDisplay">64</span>
    </div>

    
    <div class="expansion-container">
      <div class="factors-row" id="factorsRow"></div>
      <div class="count-summary" id="countSummary">
        <span style="color:#8b5cf6;">3</span> factors of <span style="color:#0284c7;">4</span>
      </div>
    </div>

  </div>

  <script>
    let currentBase = 4;
    let currentExp = 3;

    const baseButtons = document.querySelectorAll('[data-base]');
    const expButtons = document.querySelectorAll('[data-exp]');
    const baseDisplay = document.getElementById('baseDisplay');
    const expDisplay = document.getElementById('expDisplay');
    const productDisplay = document.getElementById('productDisplay');
    const factorsRow = document.getElementById('factorsRow');
    const countSummary = document.getElementById('countSummary');

    function updateDiagram() {
      // Update power expression
      baseDisplay.firstChild.textContent = currentBase;
      expDisplay.firstChild.textContent = currentExp;

      // Calculate total
      const total = Math.pow(currentBase, currentExp);
      productDisplay.textContent = total.toLocaleString();

      // Build factors row
      factorsRow.innerHTML = '';
      for (let i = 1; i <= currentExp; i++) {
        const factorGroup = document.createElement('div');
        factorGroup.className = 'factor-group';

        const badge = document.createElement('div');
        badge.className = 'count-badge';
        badge.textContent = i;

        const tile = document.createElement('div');
        tile.className = 'factor-tile';
        tile.textContent = currentBase;

        factorGroup.appendChild(badge);
        factorGroup.appendChild(tile);
        factorsRow.appendChild(factorGroup);

        if (i < currentExp) {
          const mult = document.createElement('span');
          mult.className = 'multiply-sign';
          mult.textContent = '×';
          factorsRow.appendChild(mult);
        }
      }

      // Update count text
      const factorWord = currentExp === 1 ? 'factor' : 'factors';
      countSummary.innerHTML = `<span style="color:#8b5cf6;">${currentExp}</span> ${factorWord} of <span style="color:#0284c7;">${currentBase}</span>`;
    }

    baseButtons.forEach(btn => {
      btn.addEventListener('click', () => {
        baseButtons.forEach(b => b.classList.remove('active-base'));
        btn.classList.add('active-base');
        currentBase = parseInt(btn.dataset.base, 10);
        updateDiagram();
      });
    });

    expButtons.forEach(btn => {
      btn.addEventListener('click', () => {
        expButtons.forEach(b => b.classList.remove('active-exp'));
        btn.classList.add('active-exp');
        currentExp = parseInt(btn.dataset.exp, 10);
        updateDiagram();
      });
    });

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

What is the most common mistake students make with powers?

## Watch Out: It is Not Base Times Exponent!

Never multiply the big number by the little number. $2^3$ does **not** mean $2 \times 3 = 6$.

Instead, $2^3$ means $2 \times 2 \times 2$, which equals $8$!

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Powers as Repeated Multiplication</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }

    body {
      background: #f8f9fa;
      color: #1e2945;
      padding: 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      user-select: none;
    }

    .main-wrap {
      width: 100%;
      max-width: 540px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 14px;
    }

    /* Selector Controls */
    .controls-row {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      flex-wrap: wrap;
    }

    .btn-group {
      display: inline-flex;
      background: #e5e7eb;
      padding: 3px;
      border-radius: 10px;
      gap: 2px;
    }

    .pill-btn {
      background: transparent;
      border: none;
      padding: 6px 12px;
      border-radius: 8px;
      font-size: 14px;
      font-weight: 600;
      color: #4b5563;
      cursor: pointer;
      transition: background 0.2s, color 0.2s, box-shadow 0.2s;
    }

    .pill-btn.active {
      background: #ffffff;
      color: #0284c7;
      box-shadow: 0 1px 3px rgba(0, 0, 0, 0.1);
    }

    /* Power Card */
    .display-card {
      width: 100%;
      background: #ffffff;
      border: 1px solid #e5e7eb;
      border-radius: 16px;
      padding: 18px 14px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
      box-shadow: 0 4px 12px rgba(0, 0, 0, 0.03);
    }

    /* Base and Exponent Anatomy */
    .power-expression {
      display: flex;
      align-items: baseline;
      justify-content: center;
      position: relative;
      padding-top: 18px;
    }

    .base-part {
      font-size: 52px;
      font-weight: 800;
      color: #1e2945;
      line-height: 1;
      position: relative;
    }

    .exponent-part {
      font-size: 32px;
      font-weight: 800;
      color: #0284c7;
      line-height: 1;
      transform: translateY(-22px);
      margin-left: 2px;
      position: relative;
    }

    .tag-label {
      position: absolute;
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      padding: 2px 6px;
      border-radius: 4px;
      white-space: nowrap;
    }

    .base-tag {
      top: 100%;
      left: 50%;
      transform: translateX(-50%);
      color: #64748b;
      margin-top: 4px;
    }

    .exp-tag {
      bottom: 100%;
      left: 50%;
      transform: translateX(-50%);
      background: #e0f2fe;
      color: #0284c7;
      margin-bottom: 2px;
    }

    /* Expansion Row */
    .expansion-row {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      flex-wrap: wrap;
      min-height: 48px;
    }

    .factor-pill {
      width: 40px;
      height: 40px;
      border-radius: 10px;
      background: #f0f9ff;
      border: 2px solid #38bdf8;
      color: #0369a1;
      font-size: 20px;
      font-weight: 700;
      display: flex;
      align-items: center;
      justify-content: center;
      animation: popIn 0.25s ease-out forwards;
    }

    .op-sign {
      font-size: 18px;
      font-weight: 700;
      color: #94a3b8;
    }

    .equals-sign {
      font-size: 22px;
      font-weight: 800;
      color: #64748b;
      margin: 0 4px;
    }

    .result-pill {
      min-width: 48px;
      height: 40px;
      padding: 0 12px;
      border-radius: 10px;
      background: #0284c7;
      color: #ffffff;
      font-size: 22px;
      font-weight: 800;
      display: flex;
      align-items: center;
      justify-content: center;
      box-shadow: 0 2px 8px rgba(2, 132, 199, 0.25);
    }

    /* Visual Dots Grid */
    .visual-dots-container {
      width: 100%;
      background: #f8fafc;
      border: 1px dashed #cbd5e1;
      border-radius: 12px;
      padding: 12px;
      display: flex;
      flex-wrap: wrap;
      align-items: center;
      justify-content: center;
      gap: 6px;
      max-height: 110px;
      overflow: hidden;
    }

    .dot-unit {
      width: 14px;
      height: 14px;
      background: #38bdf8;
      border-radius: 50%;
      transition: transform 0.2s;
    }

    @keyframes popIn {
      0% { transform: scale(0.5); opacity: 0; }
      100% { transform: scale(1); opacity: 1; }
    }
  </style>
</head>
<body>

  <div class="main-wrap">
    
    <div class="controls-row">
      <div class="btn-group" id="baseBtns">
        <button class="pill-btn active" onclick="setBase(2)">Base 2</button>
        <button class="pill-btn" onclick="setBase(3)">Base 3</button>
      </div>

      <div class="btn-group" id="expBtns">
        <button class="pill-btn" onclick="setExp(1)">¹</button>
        <button class="pill-btn" onclick="setExp(2)">²</button>
        <button class="pill-btn active" onclick="setExp(3)">³</button>
        <button class="pill-btn" onclick="setExp(4)">⁴</button>
      </div>
    </div>

    
    <div class="display-card">
      <div class="power-expression">
        <div class="base-part">
          <span id="baseVal">2</span>
          <div class="tag-label base-tag">Base</div>
        </div>
        <div class="exponent-part">
          <span id="expVal">3</span>
          <div class="tag-label exp-tag">Exponent</div>
        </div>
      </div>

      
      <div class="expansion-row" id="expansionContainer">
        
      </div>

      
      <div class="visual-dots-container" id="dotsContainer">
        
      </div>
    </div>
  </div>

  <script>
    let currentBase = 2;
    let currentExp = 3;

    function setBase(b) {
      currentBase = b;
      updateBaseButtons();
      render();
    }

    function setExp(e) {
      currentExp = e;
      updateExpButtons();
      render();
    }

    function updateBaseButtons() {
      const btns = document.querySelectorAll('#baseBtns .pill-btn');
      btns[0].classList.toggle('active', currentBase === 2);
      btns[1].classList.toggle('active', currentBase === 3);
    }

    function updateExpButtons() {
      const btns = document.querySelectorAll('#expBtns .pill-btn');
      btns.forEach((btn, idx) => {
        btn.classList.toggle('active', (idx + 1) === currentExp);
      });
    }

    function render() {
      document.getElementById('baseVal').textContent = currentBase;
      document.getElementById('expVal').textContent = currentExp;

      const total = Math.pow(currentBase, currentExp);
      const expansionContainer = document.getElementById('expansionContainer');
      expansionContainer.innerHTML = '';

      // Build factors: e.g. 2 × 2 × 2
      for (let i = 0; i < currentExp; i++) {
        const factor = document.createElement('div');
        factor.className = 'factor-pill';
        factor.textContent = currentBase;
        factor.style.animationDelay = (i * 0.05) + 's';
        expansionContainer.appendChild(factor);

        if (i < currentExp - 1) {
          const op = document.createElement('div');
          op.className = 'op-sign';
          op.textContent = '×';
          expansionContainer.appendChild(op);
        }
      }

      const equals = document.createElement('div');
      equals.className = 'equals-sign';
      equals.textContent = '=';
      expansionContainer.appendChild(equals);

      const res = document.createElement('div');
      res.className = 'result-pill';
      res.textContent = total;
      expansionContainer.appendChild(res);

      // Render dots
      const dotsContainer = document.getElementById('dotsContainer');
      dotsContainer.innerHTML = '';
      for (let i = 0; i < total; i++) {
        const dot = document.createElement('div');
        dot.className = 'dot-unit';
        dotsContainer.appendChild(dot);
      }
    }

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

Next time you see a tiny floating number, remember to count and multiply that base again and again!

math

Understanding Powers as Repeated Multiplication. Identify the base and exponent (index/power) and write repeated multiplication in power form.

Intro to Exponents: Repeated Multiplication

Write Multiplication Expressions Using Exponents

Powers, Roots and Indices - Third Level Maths (CfE MTH 3-06a)

Note on Curriculum Mapping: Although standard CfE nomenclature classifies 'MTH 3-06a' under Third Level (Secondary S1–S3 / typical ages 11–14) rather than Early Level (P1 / ages 4–5), this lesson plan specifically structures the core foundational concepts of MTH 3-06a. Pedagogical best practice relies on the Concrete-Pictorial-Abstract (CPA) framework: moving from tactile/geometric models of 2D squares and 3D cubes to expanded repeated multiplication strings, and ultimately to abstract base-index notation.

Understanding Powers as Repeated Multiplication

A Handy Shortcut

Base and Exponent

Writing in Power Form