Exponents & Surds

Key concepts

• a^m · a^n = a^(m+n)
• a^m / a^n = a^(m−n)
• (a^m)^n = a^(mn)
• a^(−n) = 1/aⁿ
• √a · √b = √(ab)

How it works

Apply the laws step by step. For surds, simplify and rationalise the denominator. For exponential equations, write both sides with the same base, then equate exponents. Most exponent questions come down to one of three moves: write every base as a power of a prime, factorise when the same base appears with exponents that differ by a constant, or substitute k = bˣ when the equation is a quadratic in disguise. Decide which move fits before you start writing.

Step by step

1. Same base

Rewrite each side as a power of the same base.

2. Equate exponents

Once bases match, exponents are equal.

3. Solve

Solve the resulting linear or quadratic equation.

Worked examples

Solve 2^(x+1) = 16

1. 16 = 2^4
2. x + 1 = 4 → x = 3

Answer: x = 3

Simplify (2³ · 4²) / 8²

1. Write everything as powers of 2: 4² = 2⁴ and 8² = 2⁶
2. 2³ · 2⁴ / 2⁶ = 2³⁺⁴⁻⁶

Answer: 2

Solve 3ˣ = 20, correct to two decimal places

1. Take logs: x = log 20 / log 3
2. x = 1,3010 / 0,4771

Answer: x ≈ 2,73

Simplify (x²)³ · x⁻⁴

1. Power of a power: (x²)³ = x⁶
2. Multiply by adding exponents: x⁶ · x⁻⁴ = x⁶⁺⁽⁻⁴⁾

Answer: x²

Solve 5^(x+1) + 5ˣ = 150

1. Factorise: 5ˣ(5 + 1) = 150
2. 6 · 5ˣ = 150, so 5ˣ = 25 = 5²

Answer: x = 2

Simplify √18 + √50 − √8

1. √18 = 3√2, √50 = 5√2 and √8 = 2√2
2. 3√2 + 5√2 − 2√2

Answer: 6√2

Rationalise the denominator of 4/(√5 − 1)

1. Multiply by (√5 + 1)/(√5 + 1)
2. 4(√5 + 1) / (5 − 1) = 4(√5 + 1) / 4

Answer: √5 + 1

Simplify 16^(3/4)

1. 16^(3/4) = (⁴√16)³
2. ⁴√16 = 2

Answer: 8

Common mistakes

• Adding exponents when bases are different
• Forgetting negative exponents create reciprocals
• Leaving a surd in the denominator
• Writing (a + b)² as a² + b²
• Writing (a + b)² = a² + b²: the exponent laws apply to products, not sums.
• Treating √(a + b) as √a + √b.

Frequently asked questions

Are surds in the final exam?

Yes — typically integrated into simplification or solving questions.

What is the difference between an exponent and a surd?

An exponent shows repeated multiplication, such as 2³ = 8. A surd is an irrational root, such as √2 or ∛5. Rational exponents connect the two: a^(1/n) = ⁿ√a.

When do I need logarithms with exponents?

When the two sides cannot be written with the same base, such as 3ˣ = 20. Then x = log 20 / log 3. The same method finds n in financial maths.