Further Mathematics · Lesson 3

Surds, indices & logarithms (advanced)

WAECNECO

What the syllabus asks

WAEC: Indices (laws, fractional and negative indices, exponential equations), surds (simplification, operations, rationalisation including binomial denominators, equality of surds, square roots of surds), logarithms (laws, change of base, logarithmic and exponential equations, graphs of exponential and logarithmic functions)

NECO: Indices (laws, fractional and negative indices, exponential equations), surds (simplification, operations, rationalisation including binomial denominators, equality of surds, square roots of surds), logarithms (laws, change of base, logarithmic and exponential equations, graphs of exponential and logarithmic functions)

What you will learn

Apply all the laws of indices to numerical and algebraic expressions.
Solve exponential equations by equating indices and by substitution.
Simplify, add, subtract and multiply surds.
Rationalise denominators, including binomial (a + √b) denominators.
Use equality of surds and find square roots of expressions such as 7 + 4√3.
State and apply the laws of logarithms and the change-of-base formula.
Solve logarithmic and exponential equations, checking for invalid roots, and sketch exponential and log graphs.

Key points

aᵐaⁿ = aᵐ⁺ⁿ, aᵐ/aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(m/n) = (ⁿ√a)ᵐ.
Exponential equations: same base and equate, or substitute y = aˣ (reject y ≤ 0).
Simplify surds by removing square factors; add only like surds.
(a + √b)(a − √b) = a² − b: use conjugates to rationalise.
Equality of surds: match rational parts and surd parts.
√(x + y + 2√(xy)) = √x + √y.
logₐ N = x ⇔ aˣ = N; product → sum, quotient → difference, power → multiple.
Change of base: logₐ N = log N/log a; logₐ b × log_b a = 1.
Solve log equations by combining, converting to index form and checking.
y = aˣ and y = logₐ x are reflections in y = x; through (0, 1) and (1, 0).

Start here: The laws of indices revisited

Indices are shorthand for repeated multiplication: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Every law follows from counting the factors.

Law	Reason	Example
aᵐ × aⁿ = aᵐ⁺ⁿ	m factors then n more	3⁴ × 3² = 3⁶
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	n factors cancel	5⁷ ÷ 5³ = 5⁴
(aᵐ)ⁿ = aᵐⁿ	n groups of m factors	(2³)⁴ = 2¹²
a⁰ = 1	aⁿ ÷ aⁿ = a⁰ = 1	7⁰ = 1
a⁻ⁿ = 1/aⁿ	a⁰ ÷ aⁿ	2⁻³ = 1/8
a^(1/n) = ⁿ√a	(a^(1/n))ⁿ = a	27^(1/3) = 3
a^(m/n) = (ⁿ√a)ᵐ	root first, then power	8^(2/3) = 2² = 4
(ab)ⁿ = aⁿbⁿ, (a/b)ⁿ = aⁿ/bⁿ	rearranging factors	(2x)³ = 8x³

Worked example 3.1

Evaluate (27/8)^(−2/3).

Negative index: flip the fraction, giving (8/27)^(2/3).
Take the cube root first, because it keeps the numbers small: ∛(8/27) = 2/3.
Square: (2/3)² = 4/9.

4/9

Worked example 3.2

Evaluate 16^(3/4) × 4^(−1/2).

16^(3/4) = (⁴√16)³ = 2³ = 8.
4^(−1/2) = 1/√4 = 1/2.
8 × 1/2 = 4.

⚠️ Common mistake: thinking a negative index gives a negative number. 2⁻³ = 1/8, which is positive.

Try it yourself. Evaluate (0.125)^(−1/3).

Show solution

0.125 = 1/8, so (1/8)^(−1/3) = 8^(1/3) = 2.

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Practice questions

1. Evaluate (64)^(−2/3).

Show worked answer

1/(∛64)² = 1/16.

2. Solve 8^(x−1) = 4^(x+1).

Show worked answer

3x − 3 = 2x + 2, x = 5.

3. Simplify 3√8 + √32 − √18.

Show worked answer

6√2 + 4√2 − 3√2 = 7√2.

4. Rationalise (√3 + 1)/(√3 − 1).

Show worked answer

(√3 + 1)²/2 = (4 + 2√3)/2 = 2 + √3.

5. Find √(8 + 2√15).

Show worked answer

x + y = 8, xy = 15: √5 + √3.

Key words

Index (exponent): The power to which a base is raised, e.g. 3 in 2³.
Base: The number being raised to a power.
Exponential equation: An equation with the unknown in the index, e.g. 2ˣ = 32.
Surd: An irrational root of a rational number, e.g. √2, ∛5.
Like surds: Surds with the same irrational part, e.g. 3√5 and 7√5.
Conjugate: The expression a − √b paired with a + √b; their product is rational.
Rationalise: To remove surds from a denominator.
Logarithm: logₐ N is the power to which a must be raised to give N.
Common logarithm: Logarithm to base 10, written log or lg.
Natural logarithm: Logarithm to base e (≈ 2.718), written ln.
Characteristic: The whole-number part of a common logarithm.
Mantissa: The decimal part of a common logarithm.

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