Grade 10 Exponents and Exponential Equations

Key concepts

• aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ and (aᵐ)ⁿ = aᵐⁿ
• (ab)ⁿ = aⁿbⁿ, a⁰ = 1 and a⁻ⁿ = 1/aⁿ (a ≠ 0)
• Writing numbers as powers of primes: 8 = 2³, 27 = 3³, 125 = 5³
• Solving exponential equations by making the bases equal
• Factorising expressions with exponents, e.g. 2ˣ⁺² − 2ˣ = 2ˣ(4 − 1)

Worked examples

Simplify: (2⁵ × 2⁻²) ÷ 2²

Add and subtract the exponents: 5 + (−2) − 2 = 1, so the answer is 2¹ = 2.

Solve for x: 3ˣ⁺¹ = 27

Write 27 as 3³: 3ˣ⁺¹ = 3³. The bases are equal, so x + 1 = 3 and x = 2.

Simplify: (2ˣ⁺² − 2ˣ) ÷ (3 · 2ˣ)

Factorise the numerator: 2ˣ(2² − 1) = 2ˣ × 3. Divide by 3 · 2ˣ to get 1.

Simplify without a calculator: (3² × 9) ÷ 27

Write everything as powers of 3: 9 = 3² and 27 = 3³. So (3² × 3²) ÷ 3³ = 3²⁺²⁻³ = 3¹ = 3.

Common mistakes

• Multiplying the bases when multiplying powers: 2³ × 2⁴ is 2⁷, not 4⁷.
• Thinking a negative exponent makes the number negative: 2⁻³ = 1/8, not −8.
• Adding exponents of different bases, e.g. 2³ × 3² is not 6⁵.
• Cancelling terms in a sum: (2ˣ + 2) ÷ 2 is not 2ˣ.

Exam tips

• Break every base down to primes before you apply any law.
• When terms are added or subtracted, factorise first; the laws only work on products and quotients.
• Check an exponential equation by substituting your answer back in.

Frequently asked questions

Do Grade 10 learners use rational exponents like 8^(2/3)?

Rational exponents and surd equations are Grade 11 work in CAPS. Grade 10 focuses on integer exponents and simple exponential equations.

Why is a⁰ = 1?

Because aⁿ ÷ aⁿ = 1, and by the subtraction law aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. So a⁰ must equal 1 for any a ≠ 0.

How do I solve 2ˣ = 10?

10 is not a power of 2, so this needs logarithms, which are Grade 12 work. Grade 10 questions always use numbers you can write with the same base.