Exponents & Surds for Matric Maths

Laws of exponents, surd manipulation and exponential equations for NSC Paper 1. CAPS-aligned with AI step-by-step help.

Exponents and surds underpin algebra, functions and finance. Master the laws of exponents, rationalising denominators and solving exponential equations. In an exam, write each exponent law on the line where you use it and keep surd answers exact unless the question explicitly asks for a decimal approximation; this protects method marks and avoids premature rounding.

Key concepts

  • aᵐ · aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ
  • Rationalising surd denominators
  • Solving exponential equations using a common base
  • Rational exponents: a^(m/n) = ⁿ√(aᵐ)
  • Factorising out a common power, e.g. 2^(x+2) − 2ˣ = 2ˣ(4 − 1)
  • Exponential equations that become quadratics with k = 2ˣ
  • Writing bases as products of prime factors to simplify

Worked examples

Solve 2^(x+1) = 32.

32 = 2⁵, so x + 1 = 5 and x = 4.

Simplify √12 + √27

√12 = 2√3 and √27 = 3√3, so the sum is 5√3.

Solve 9ˣ = 27

Write both sides as powers of 3: 3²ˣ = 3³, so 2x = 3 and x = 3/2.

Simplify 8^(2/3) without a calculator.

The denominator of the exponent is the root and the numerator is the power: 8^(2/3) = (∛8)² = 2² = 4. Taking the root first keeps the numbers small.

Solve 3^(x+2) − 3ˣ = 72.

Factorise: 3ˣ(3² − 1) = 72, so 8 · 3ˣ = 72 and 3ˣ = 9 = 3². Therefore x = 2.

Solve 2^(2x) − 6·2ˣ + 8 = 0.

Let k = 2ˣ: k² − 6k + 8 = 0, so (k − 2)(k − 4) = 0. Then 2ˣ = 2 or 2ˣ = 4, so x = 1 or x = 2.

Simplify (2ˣ · 4^(x+1)) / 8ˣ.

Write every base as a power of 2: 2ˣ · 2^(2x+2) / 2^(3x) = 2^(x + 2x + 2 − 3x) = 2² = 4.

Common mistakes

  • Adding bases instead of adding exponents.
  • Forgetting that √a · √a = a (not a²).
  • Writing 2ˣ + 2ˣ = 4ˣ instead of 2 · 2ˣ = 2^(x+1).
  • Cancelling terms instead of factors in fractions with exponents.

Exam tips

  • Always rewrite to a common base before solving exponential equations.
  • Factorise when the bases are the same but the exponents differ by a constant.
  • Check solutions by substituting back: 2ˣ = −4 has no real solution because 2ˣ is always positive.

In past papers

  • NSC Nov 2024 Paper 1 Q1.5

Frequently asked questions

Are logarithms in the CAPS syllabus?

Yes, in two places: the logarithmic function y = log_b x as the inverse of y = bˣ, and using logarithms to solve for n in financial maths and geometric sequences. Know the definition y = log_b x ⇔ x = bʸ.

How do I solve an exponential equation with different bases?

Write both sides as powers of the same prime base and equate the exponents. If that is not possible, isolate the power and use logarithms.

Why should I leave answers in surd form?

An exact answer such as 2√3 has no rounding error. Convert to a decimal only when the question asks for a rounded answer.

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