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Imagine a computer file doubling in size every hour. If it doubles 3 times this morning and 4 more times tonight, how many total doublings took place?

In math, repeated multiplication uses a **base** (the value being multiplied) and an **exponent** (the number of times it multiplies), written as $2^3 \cdot 2^4$.

In 1637, French mathematician René Descartes introduced raised superscript exponents so mathematicians wouldn't have to keep writing out long chains like $x \cdot x \cdot x$.

So how do we combine these powers when multiplying them together?

When multiplying terms with the **same base**, keep the base and add the exponents: $a^m \cdot a^n = a^{m+n}$.

This rule works because writing out each power simply joins all the repeated factors into one single, longer chain.

This rule relies on the **Same-Base Condition**: powers can only be combined if their bases are identical. Expressions with different bases, such as $x^3 \cdot y^4$ or $2^3 \cdot 3^4$, cannot have their exponents added directly.

A crucial requirement is the **same-base condition**: the rule applies exclusively to expressions with identical bases.

Powers with different bases (such as $x^3 \cdot y^4$ or $2^3 \cdot 3^4$) cannot have their exponents added directly.

Be careful with **hidden exponents**: a variable or number written without an explicit exponent has an implicit exponent of $1$ ($x = x^1$).

This means $x \cdot x^4 = x^1 \cdot x^4 = x^{1+4} = x^5$.

Also watch out for **hidden exponents**: a variable or number written without an exponent has an implicit power of $1$ ($x = x^1$), meaning $x \cdot x^4 = x^1 \cdot x^4 = x^{1+4} = x^5$.

<!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>Product of Powers Rule</title>
  <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.css">
  <script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.8/katex.min.js"></script>
  <style>
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    body {
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      flex-direction: column;
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      justify-content: center;
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    .wrapper {
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      max-width: 540px;
      display: flex;
      flex-direction: column;
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      gap: 14px;
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    /* Stepper Tabs */
    .tabs {
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      padding: 4px;
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    /* Display Stage */
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      background: #ffffff;
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      border: 1px solid #e2e8f0;
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      display: flex;
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    /* Base selector / example chooser */
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    /* Factor Groups */
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    .group-a {
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    .token {
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    .token-a {
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  </style>
</head>
<body>
  <div class="wrapper">
    
    <div class="tabs">
      <button class="tab-btn active" onclick="setStep(0)">1. Powers</button>
      <button class="tab-btn" onclick="setStep(1)">2. Expand</button>
      <button class="tab-btn" onclick="setStep(2)">3. Add Chain</button>
    </div>

    
    <div class="stage">
      <div class="expr-hero" id="mathHero"></div>
      
      <div class="chains-container" id="chainView"></div>

      <div class="bracket-label" id="labelView"></div>
    </div>

    
    <div class="controls-row">
      <button class="btn-nav" id="prevBtn" onclick="prevStep()">← Prev</button>
      <div class="dot-indicators">
        <span class="dot active" onclick="setStep(0)"></span>
        <span class="dot" onclick="setStep(1)"></span>
        <span class="dot" onclick="setStep(2)"></span>
      </div>
      <button class="btn-nav" id="nextBtn" onclick="nextStep()">Next →</button>
    </div>
  </div>

  <script>
    let currentStep = 0;
    const base = 'x';
    const m = 3;
    const n = 2;

    function renderTokens(count, type) {
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    function update() {
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      document.getElementById('prevBtn').disabled = currentStep === 0;
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        katex.render(`${base}^{${m}} \\cdot ${base}^{${n}}`, hero);
        chain.innerHTML = `
          <div class="factor-group group-a">${renderTokens(m, 'a')}</div>
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          <div class="factor-group group-b">${renderTokens(n, 'b')}</div>
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          <span class="badge badge-a">${m} factors</span>
          <span class="badge badge-b">${n} factors</span>
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          <div class="factor-group group-a">${renderTokens(m, 'a')}</div>
          <span class="dot-op" style="font-size:22px;">•</span>
          <div class="factor-group group-b">${renderTokens(n, 'b')}</div>
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          <span class="badge badge-a">3 factors</span>
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</html>

When simplifying algebraic terms, practice **separating coefficients** from their variable bases.

In an expression like $(3x^4)(5x^{-2})$, multiply the coefficients ($3 \times 5 = 15$) and add the exponents of matching variable bases ($x^{4+(-2)} = x^2$) to get $15x^2$.

What happens when an exponent is zero or negative? Does adding exponents still work?

Every time an exponent decreases by $1$, we divide by the base.

Continuing this division pattern below $a^1$ reveals that any non-zero base to the power of zero equals $1$ ($a^0 = 1$), and a negative exponent indicates repeated division: $a^{-n} = \frac{1}{a^n}$.

<!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>Product of Powers with Negative Exponents</title>
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css">
<script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.js"></script>
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    background: transparent;
    color: #1e2945;
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    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 100%;
    padding: 16px 12px;
  }
  .stage {
    width: 100%;
    max-width: 540px;
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 16px;
  }
  .controls {
    display: flex;
    gap: 8px;
    background: #eef2f6;
    padding: 4px;
    border-radius: 24px;
  }
  .tab-btn {
    border: none;
    background: transparent;
    padding: 6px 14px;
    font-size: 14px;
    font-weight: 600;
    color: #64748b;
    border-radius: 20px;
    cursor: pointer;
    transition: all 0.25s ease;
  }
  .tab-btn.active {
    background: #ffffff;
    color: #0284c7;
    box-shadow: 0 2px 6px rgba(0,0,0,0.08);
  }
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    width: 100%;
    min-height: 240px;
    background: #ffffff;
    border: 1px solid #e2e8f0;
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    align-items: center;
    justify-content: center;
    padding: 20px 16px;
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  }
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    display: flex;
    align-items: center;
    justify-content: center;
    font-size: 24px;
    min-height: 48px;
    transition: opacity 0.3s ease;
  }
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    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 8px;
    margin: 12px 0;
  }
  .factor-row {
    display: flex;
    gap: 8px;
    align-items: center;
    justify-content: center;
    flex-wrap: wrap;
  }
  .fraction-line {
    width: 100%;
    min-width: 220px;
    height: 2px;
    background: #1e2945;
    border-radius: 1px;
  }
  .factor {
    display: inline-flex;
    align-items: center;
    justify-content: center;
    width: 38px;
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    font-family: KaTeX_Math, serif;
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  .factor.pos { background: #e0f2fe; color: #0369a1; }
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    border: 1px dashed #cbd5e1;
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</style>
</head>
<body>

<div class="stage">
  <div class="controls">
    <button class="tab-btn active" id="tab0">y⁵ · y⁻³</button>
    <button class="tab-btn" id="tab1">a⁴ · a⁻²</button>
    <button class="tab-btn" id="tab2">x³ · x⁻⁴</button>
  </div>

  <div class="display-box" id="displayBox">
    
  </div>

  <div class="step-bar">
    <button class="step-btn" id="prevBtn" aria-label="Previous">◀</button>
    <div class="dots" id="dotsContainer"></div>
    <button class="step-btn" id="nextBtn" aria-label="Next">▶</button>
    <button class="step-btn" id="playBtn" style="margin-left: 6px;" aria-label="Play/Pause">⏸</button>
  </div>
</div>

<script>
const examples = [
  { base: 'y', pos: 5, neg: 3 },
  { base: 'a', pos: 4, neg: 2 },
  { base: 'x', pos: 3, neg: 4 }
];

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let curStep = 0;
const totalSteps = 4;
let timer = null;
let isPlaying = true;

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function nextStep() {
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function goToStep(s) {
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document.getElementById('prevBtn').addEventListener('click', () => { prevStep(); stopAuto(); });
playBtn.addEventListener('click', () => {
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});

[0, 1, 2].forEach(i => {
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    curStep = 0;
    render();
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});

// Init
render();
startAuto();
</script>

</body>
</html>

Because a negative exponent represents dividing out factors, adding a negative exponent naturally cancels factors out: $y^5 \cdot y^{-3} = y^{5 + (-3)} = y^2$.

This demonstrates the **Zero Exponent Identity**: multiplying by a zero exponent acts as multiplying by the identity $1$ ($a^m \cdot a^0 = a^{m+0} = a^m$), consistent with $a^0 = 1$ for any $a \neq 0$.

How do we tackle larger expressions that mix multiple variables, coefficients, and negative signs?

Group coefficients together and group matching variable bases together before applying the exponent addition rule.

Let's simplify the product $(2a^{-4}b^3)(5a^6b^{-3})$ step by step.

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    <button class="step-btn active" onclick="setStep(0)">1. Problem</button>
    <button class="step-btn" onclick="setStep(1)">2. Group</button>
    <button class="step-btn" onclick="setStep(2)">3. Add Exponents</button>
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A common mistake is multiplying the bases instead of keeping the base the same—$2^3 \cdot 2^4$ is $2^7$, never $4^7$.

Another frequent error is multiplying the exponents instead of adding them; remember that $x^3 \cdot x^2$ creates $5$ total factors of $x$, not $6$.

As long as the bases match, you can confidently combine any number of positive, negative, or zero powers by simply adding their exponents.

math

Product of Powers Rule with Integer Exponents. Multiply expressions with the same base by adding exponents, including negative and zero exponents.

Multiplying & dividing powers (integer exponents)

Multiply powers: integer bases

Laws of Indices: Steps, Examples & Practice

To build deep conceptual understanding aligned with Australian Curriculum AC9M9A01, transition students through concrete-representational-abstract (CRA) phases: (1) Expand positive powers into factor strings to rediscover index addition; (2) Introduce negative powers via fraction cancellation models to show why integer subtraction/addition naturally merges; (3) Explicitly isolate coefficients from exponents using color-coding to prevent cross-operation interference (multiplying bases vs. adding powers); (4) Culminate in mixed-sign multivariate expressions containing hidden powers ($x^1$) and zero powers ($y^0$).

Product of Powers Rule with Integer Exponents

The Product of Powers Rule