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Grade 8 · Exponents · 10 min read

The Laws of Exponents (Grade 8)

The five exponent laws in the Grade 8 CAPS syllabus — multiplying, dividing, a power of a power, a power of a product, and the zero exponent — each explained from first principles with worked examples, a worksheet and a quiz.

Rather than memorising five rules, notice that every one of them comes from simply writing the powers out in full. Do that once for each law and you will never need to guess which rule applies.

1Law 1: multiplying — add the exponents

2³ = 8 2² = 4 2¹ = 2 2⁰ = 1 2⁻¹ = ½ 2⁻² = ¼ each step down divides by 2 — which is why 2⁰ = 1
Step down the ladder and each row divides by the base — this is where the zero exponent comes from.
am × an = am+n
Worked ExampleSimplify: 23 × 24
  1. 1
    Write both out: (2×2×2) × (2×2×2×2).
    This is why the law works.
  2. 2
    That is seven 2's multiplied together = 27.
    Count the factors: 3 + 4 = 7.
  3. 3
    So 23 × 24 = 27.
    Same base → add the exponents.
⚠️

This law only works when the bases are the same. 23 × 34 cannot be combined this way.

2Law 2: dividing — subtract the exponents

am ÷ an = am−n
Worked ExampleSimplify: 56 ÷ 52
  1. 1
    Write as a fraction: (5×5×5×5×5×5) / (5×5).
    Division is cancelling.
  2. 2
    Two 5's cancel top and bottom, leaving four 5's.
    6 − 2 = 4.
  3. 3
    So 56 ÷ 52 = 54.
    Same base → subtract the exponents.

3Law 3: a power of a power — multiply the exponents

(am)n = am×n
Worked ExampleSimplify: (32)4
  1. 1
    (32)4 means 32 × 32 × 32 × 32.
    The outer exponent says how many copies.
  2. 2
    Adding exponents: 2+2+2+2 = 8.
    Which is the same as 2 × 4.
  3. 3
    So (32)4 = 38.
    Power of a power → multiply.

4Law 4: a power of a product

(a × t)n = an × tn
Worked ExampleSimplify: (2x)3
  1. 1
    (2x)3 = 2x × 2x × 2x.
    Everything inside the bracket is cubed.
  2. 2
    = (2×2×2) × (x×x×x) = 8x3.
    Group the numbers and the letters.
⚠️

Watch the bracket: (2x)3 = 8x3, but 2x3 = 2 × x3. Without a bracket only the x is cubed.

5Law 5: the zero exponent

a0 = 1  (for any a ≠ 0)
Worked ExampleWhy does 50 = 1?
  1. 1
    By Law 2: 53 ÷ 53 = 53−3 = 50.
    Subtract the exponents.
  2. 2
    But 53 ÷ 53 = 125 ÷ 125 = 1.
    Anything divided by itself is 1.
  3. 3
    So 50 must equal 1.
    Both routes must give the same answer.

6Putting the laws together

Worked ExampleSimplify: (23 × 22) ÷ 24
  1. 1
    Brackets first (Law 1): 23 × 22 = 25.
    Same base, add: 3 + 2 = 5.
  2. 2
    Now divide (Law 2): 25 ÷ 24 = 21.
    Subtract: 5 − 4 = 1.
  3. 3
    Answer: 21 = 2.
    Always give the final value where possible.

Practice exercises

Work each one out, then click to reveal the answer.

📄 Worksheet✅ Memo📝 Exam paper🗝️ Marking guide
  1. 1
    Simplify: 34 × 33
    Show answer ▾
    Add: 37
  2. 2
    Simplify: 78 ÷ 75
    Show answer ▾
    Subtract: 73
  3. 3
    Simplify: (53)2
    Show answer ▾
    Multiply: 56
  4. 4
    Simplify: (3y)4
    Show answer ▾
    81y4
  5. 5
    Calculate: 120
    Show answer ▾
    1
  6. 6
    Simplify: x5 × x2
    Show answer ▾
    x7
  7. 7
    Simplify: (22 × 25) ÷ 23
    Show answer ▾
    27 ÷ 23 = 24 = 16
  8. 8
    Simplify: (a3)2 × a
    Show answer ▾
    a6 × a1 = a7
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Quick Quiz

5 quick questions on what you just read. Take it when you feel ready.

Now practise it

Download Grade 8 past papers and worksheets on this topic.

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➕
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