Exponent Laws & Exponential Equations — Grade 11 (CAPS)

The exponent laws including rational exponents, simplifying expressions with different bases, and solving exponential equations by making the bases the same or by factorising.

Watch these lessons

1The laws

a^m × aⁿ = a^{m + n} · a^m/aⁿ = a^{m − n} · (a^m)ⁿ = a^mn
a⁰ = 1 · a⁻ⁿ = 1/aⁿ · a^m/n = ⁿ√a^m
for a base a > 0 (so it is never 0), and n > 0 in the root

💡

A negative exponent is an instruction to move, not to make negative. 2⁻³ = 1/8, never −8.

2Rational exponents

Worked ExampleSimplify 16^3/4

1
The denominator 4 is the root: the FOURTH root of 16 = 2.
Take the root first — the numbers stay small.
2
The numerator 3 is the power: 2³ = 8.
3
= 8.

3Simplifying with different bases

Worked ExampleSimplify (9ⁿ × 3^{n + 1})/27ⁿ

1
Write every base as a power of 3: 9 = 3², 27 = 3³.
You can only add exponents when the bases match.
2
= (3²ⁿ × 3^{n + 1})/3³ⁿ.
(3²)ⁿ = 3²ⁿ.
3
= 3^{2n + n + 1 − 3n}.
Add on top, subtract below.
4
= 3¹ = 3.
2n + n − 3n = 0.

4Exponential equations: same base

Worked ExampleSolve 3^{x + 2} = 81

1
81 = 3⁴.
Write both sides with base 3.
2
3^{x + 2} = 3⁴.
3
x + 2 = 4, so x = 2.
Equal bases ⇒ equal exponents — the base 3 is positive and not 1, which is what makes that step legal.

5Exponential equations: factorising

Worked ExampleSolve 2^2x − 5(2^x) + 4 = 0

1
Let k = 2^x. Then 2^2x = k².
2^2x = (2^x)² — this substitution is the whole trick.
2
k² − 5k + 4 = 0 → (k − 1)(k − 4) = 0.
3
k = 1 or k = 4, so 2^x = 1 or 2^x = 4.
Substitute back — never leave the answer as k.
4
x = 0 or x = 2.
2⁰ = 1 and 2² = 4.

⚠️

2^x can never be zero or negative. If a substitution gives k = −3, reject it and say why.

Practice exercises

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

📄 Worksheet ✅ Memo 📝 Exam paper 🗝️ Marking guide

1
Simplify 16^3/4
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(4th root of 16)³ = 2³ = 8
2
Simplify 2⁻³
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1/8
3
Simplify (9ⁿ × 3^{n + 1})/27ⁿ
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3^{2n + n + 1 − 3n} = 3¹ = 3
4
Solve for x: 3^{x + 2} = 81
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3^{x + 2} = 3⁴ → x = 2
5
Solve for x: 5^{2x − 1} = 125
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2x − 1 = 3 → x = 2
6
Solve for x: 2^2x − 5(2^x) + 4 = 0
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Let k = 2^x: (k − 1)(k − 4) = 0 → x = 0 or x = 2
7
Solve for x: 3^2x − 4(3^x) + 3 = 0
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(k − 1)(k − 3) = 0 → x = 0 or x = 1
8
Simplify 8^2/3
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(cube root of 8)² = 2² = 4