The Laws of Exponents — Grade 9 Maths (CAPS)

Every exponent law with its own worked example — multiplying, dividing, powers of powers and products, zero and negative exponents — plus combined exam-style questions and a full worksheet with answers.

Also known as: laws of indices · index laws · exponent rules

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Grade 9–10 · Video lessonExponential Expressions Examples

Exponents (indices) are shorthand for repeated multiplication: x⁴ means x × x × x × x. Every 'law' below is just that definition applied carefully — so if you forget a rule, you can rebuild it in seconds. This guide covers every exponent law and the ways a Grade 9 exam combines them.

In x⁴, the base is x and the exponent is 4. The laws only work when the bases are the same.

1Law 1: Multiplying (same base): ADD the exponents

x^a × x^b = x^{a + b}

Worked ExampleExample 1: simplify 2x³ · 5x⁴

1
Multiply the numbers: 2 × 5 = 10.
Coefficients multiply normally.
2
Add the exponents of x: x³ · x⁴ = x⁷.
Same base, so add: 3 + 4 = 7.
3
Answer: 10x⁷.
Combine the number and the variable part.

2Law 2: Dividing (same base): SUBTRACT the exponents

x^a ÷ x^b = x^{a − b} (x ≠ 0)

Worked ExampleExample 2: simplify 12x⁵/3x²

1
Divide the numbers: 12 ÷ 3 = 4.
Coefficients divide normally.
2
Subtract the exponents of x: x⁵ ÷ x² = x³.
Same base, so subtract: 5 − 2 = 3.
3
Answer: 4x³.
Combine the parts.

3Law 3: Power of a power: MULTIPLY the exponents

(x^a)^b = x^{a × b}

Worked ExampleExample 3: simplify (x³)⁴

1
Multiply the exponents: 3 × 4 = 12.
A power raised to a power multiplies.
2
Answer: x¹².
The base stays x.

4Law 4: Power of a product: give EVERY factor the power

(ab)ⁿ = aⁿbⁿ

Worked ExampleExample 4: simplify (2x²)³

1
Apply the power 3 to every factor inside: 2 and x².
Both the number and the variable are raised to 3.
2
2³ = 8 and (x²)³ = x⁶.
Cube the 2; multiply exponents for the x.
3
Answer: 8x⁶.
The coefficient 2 becomes 8 — don't forget it.

5Law 5: Zero exponent: the answer is 1

Anything (except 0) to the power 0 equals 1. Why? Because x³ ÷ x³ = x⁰, but any non-zero number divided by itself is 1.

Worked ExampleExample 5: evaluate 3x⁰ and (3x)⁰, where x ≠ 0

1
In 3x⁰, only the x has the power 0, so x⁰ = 1 → 3 × 1 = 3.
The power sits on x only, not the 3.
2
In (3x)⁰, the whole bracket is to the power 0 → 1.
Brackets mean everything inside gets the power.

6Law 6: Negative exponents: 'one over'

A negative power means take the reciprocal — it does not make the answer negative.

x⁻ⁿ = 1/xⁿ (x ≠ 0)

Worked ExampleExample 6: write 2x⁻³ with a positive exponent

1
The x⁻³ moves to the denominator as x³.
Negative exponent means 'one over'.
2
The 2 stays on top (its exponent is +1).
Only the x has the negative power.
3
Answer: 2/x³.
= 2/x³, a positive-exponent form.

7Putting it together — combined exam questions

Real exam questions mix several laws. Work from the inside out: brackets first, then multiply/divide.

Worked ExampleExample 7: simplify (2x²)³ × x⁻¹

1
Bracket first (Law 4): (2x²)³ = 2³x⁶ = 8x⁶.
Power of a product — cube the 2 as well.
2
Multiply by x⁻¹ (Law 1): 8x⁶ × x⁻¹ = 8x^{6 − 1}.
Same base, add exponents: 6 + (−1) = 5.
3
Answer: 8x⁵.
A single positive-exponent term.

Worked ExampleExample 8: simplify 6x⁴y²/2x²y⁵

1
Numbers: 6 ÷ 2 = 3.
Divide the coefficients.
2
x: x⁴ ÷ x² = x². y: y² ÷ y⁵ = y⁻³.
Subtract exponents for each base separately.
3
Write with positive exponents: 3x²/y³.
y⁻³ becomes 1/y³, so it drops to the denominator.

8Scientific notation (Grade 8–9)

Scientific notation writes a number as a value of at least 1 but less than 10, times a power of 10 — useful for very large or very small numbers.

Worked ExampleExample 9: write 45 000 in scientific notation

1
Place the decimal comma after the first non-zero digit: 4,5.
The value must be at least 1 and less than 10.
2
Count how many places the decimal moved: 4 places → ×10⁴.
45 000 → 4,5 needs 4 jumps.
3
Answer: 4,5 × 10⁴.
Large number → positive power of 10.

⚠️

Two classic slips: (1) forgetting the coefficient gets the power — (2x)³ is 8x³, not 2x³; (2) thinking a negative exponent makes the answer negative — it makes a fraction, not a minus.

💡

Numbers and variables are handled separately. Deal with the coefficients (multiply/divide the numbers) and the exponents (add/subtract/multiply) as two separate jobs, then recombine.

Every law above is examined from Grade 8 right through to matric. Drill the worksheet until you can spot which law applies at a glance — exponents are among the fastest, most reliable marks in the paper.

Practice exercises

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

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Simplify x⁴ · x³ (Law 1)
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Add exponents → x⁷
2
Simplify 3a² · 4a⁵ (Law 1)
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3 × 4 = 12, add exponents → 12a⁷
3
Simplify m⁸ ÷ m³ (Law 2)
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Subtract exponents → m⁵
4
Simplify 20x⁶/5x² (Law 2)
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20 ÷ 5 = 4, subtract exponents → 4x⁴
5
Simplify (a³)² (Law 3)
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Multiply exponents → a⁶
6
Simplify (2y⁴)³ (Law 4)
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2³ = 8, and the exponents multiply: 4 × 3 = 12 → 8y¹²
7
Simplify (3x²)² (Law 4)
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3² = 9 → 9x⁴
8
Evaluate 7x⁰, where x ≠ 0 (Law 5)
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x⁰ = 1 → 7
9
Evaluate (5y)⁰, where y ≠ 0 (Law 5)
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Whole bracket to power 0 → 1
10
Write 4x⁻² with a positive exponent (Law 6)
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4/x²
11
Simplify (2x³)² · x (combined)
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(2x³)² = 4x⁶, ×x → 4x⁷
12
Simplify 12a⁵b²/4a²b⁶ (combined)
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12 ÷ 4 = 3, a^{5 − 2} = a³, b^{2 − 6} = b⁻⁴ → 3a³/b⁴
13
Write 3 200 000 in scientific notation
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3,2 × 10⁶
14
Write 0,0007 in scientific notation
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Small number → negative power → 7 × 10⁻⁴