Exponent Laws & Exponential Equations (Grade 11)

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

1The laws

a^m × aⁿ = a^m+n · a^maⁿ = a^m−n · (a^m)ⁿ = a^mn
a⁰ = 1 · a⁻ⁿ = 1aⁿ · a^mn = a^m (n-th root)

💡

A negative exponent is an instruction to move, not to make negative. 2⁻³ = 18, never −8.

2Rational exponents

Worked ExampleSimplify 16³⁴

1
The denominator 4 is the root: 16 (4th root) = 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 × 3ⁿ⁺¹27ⁿ

1
Write every base as a power of 3: 9 = 3², 27 = 3³.
You can only add exponents when the bases match.
2
= 3²ⁿ × 3ⁿ⁺¹3³ⁿ.
(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.

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 16³⁴
Show answer ▾Hide answer ▴
(4th root of 16)³ = 2³ = 8
2
Simplify 2⁻³
Show answer ▾Hide answer ▴
18
3
Simplify 9ⁿ × 3ⁿ⁺¹27ⁿ
Show answer ▾Hide answer ▴
3^2n+n+1−3n = 3¹ = 3
4
Solve for x: 3^x+2 = 81
Show answer ▾Hide answer ▴
3^x+2 = 3⁴ → x = 2
5
Solve for x: 5^2x−1 = 125
Show answer ▾Hide answer ▴
2x − 1 = 3 → x = 2
6
Solve for x: 2^2x − 5(2^x) + 4 = 0
Show answer ▾Hide answer ▴
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
Show answer ▾Hide answer ▴
(k−1)(k−3) = 0 → x = 0 or x = 1
8
Simplify 8²³
Show answer ▾Hide answer ▴
(cube root of 8)² = 2² = 4