Indices and Surds

Indices (exponents) and surds (irrational roots) are essential topics in quantitative reasoning. Indices deal with repeated multiplication and the rules for combining powers, while surds deal with roots that cannot be simplified to rational numbers. Both topics are frequently tested in Post-UTME and form the foundation for algebraic manipulation.

The Laws of Indices

An index (or exponent/power) tells us how many times a number is multiplied by itself: aⁿ = a × a × … (n times).

The Laws

Law	Statement	Example
Product of Powers	aᵐ × aⁿ = aᵐ⁺ⁿ	3² × 3⁴ = 3⁶ = 729
Quotient of Powers	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	5⁶ ÷ 5² = 5⁴ = 625
Power of a Power	(aᵐ)ⁿ = aᵐⁿ	(2³)⁴ = 2¹² = 4096
Power of a Product	(ab)ⁿ = aⁿbⁿ	(3×4)² = 9×16 = 144
Power of a Quotient	(a/b)ⁿ = aⁿ/bⁿ	(3/4)² = 9/16
Zero Index	a⁰ = 1 (a ≠ 0)	5⁰ = 1
Negative Index	a⁻ⁿ = 1/aⁿ	2⁻³ = 1/8

Fractional Indices

a^(m/n) = (ⁿ√a)ᵐ = (a^(1/n))^m

• a^(1/2) = √a; a^(1/3) = ∛a

Example: 8^(2/3) = (8^(1/3))² = 2² = 4; 16^(3/4) = (16^(1/4))³ = 2³ = 8

Evaluating Indices

Example: Simplify 2³ × 2⁴ ÷ 2⁵ = 2³⁺⁴⁻⁵ = 2² = 4

Example: Simplify (3²)³ × 3⁰ ÷ 3⁴ = 3⁶ × 1 ÷ 3⁴ = 3⁶⁻⁴ = 3² = 9

Example: Simplify (2⁻³ × 2⁵) ÷ 2⁻² = 2² ÷ 2⁻² = 2²⁻⁽⁻²⁾ = 2⁴ = 16

Standard Form (Scientific Notation)

A number in standard form is written as: a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.

Converting: 5,300,000 = 5.3 × 10⁶; 0.00042 = 4.2 × 10⁻⁴

Operations: (a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ Example: (3 × 10⁸) × (2 × 10⁵) = 6 × 10¹³

Surds

A surd is an irrational root of a rational number — one that cannot be simplified to a whole number. Common surds include √2, √3, √5.

Simplifying Surds

Rule: √(a × b) = √a × √b

• √50 = √(25 × 2) = √25 × √2 = 5√2
• √72 = √(36 × 2) = 6√2

Operations on Surds

Addition/Subtraction: Only like surds (same radicand) can be added.

• 3√2 + 5√2 = (3+5)√2 = 8√2
• 3√2 + 5√3: Cannot be combined

Multiplication: √a × √b = √(ab)

• √3 × √5 = √15; 3√2 × 5√2 = 3 × 5 × 2 = 30 (since √2 × √2 = 2)

Rationalizing the Denominator

Type 1: Denominator is √a → Multiply by √a/√a

• 1/√3 = (1×√3)/(√3×√3) = √3/3

Type 2: Denominator is a + √b (two-term denominator) → Multiply by the conjugate (a − √b)

• 1/(2 + √3) = (2 − √3)/[(2+√3)(2−√3)] = (2 − √3)/(4−3) = 2 − √3

Example: Rationalize 5/(3 − √2) = 5(3 + √2)/[(3−√2)(3+√2)] = 5(3+√2)/7

Key Post-UTME Exam Facts

• Indices: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1; a⁻ⁿ = 1/aⁿ
• Fractional indices: a^(m/n) = (ⁿ√a)ᵐ
• Surd multiplication: √a × √b = √(ab)
• Adding surds: Only combine like surds
• Rationalizing: For 1/(√a+√b) → multiply by conjugate (√a−√b)
• Standard form: a × 10ⁿ where 1 ≤ a < 10
• ⚡ Exam tip: √a × √a = a; (√a)² = a; 2√3 + 5√3 = 7√3, but 2√3 + 5√2 cannot be combined

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🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

• Index Laws: Product = add exponents; Division = subtract exponents; Power of power = multiply exponents
• Zero index: a⁰ = 1 (a ≠ 0)
• Negative index: a⁻ⁿ = 1/aⁿ
• Fractional index: a^(1/n) = n-th root; a^(m/n) = (ⁿ√a)ᵐ
• Simplifying surds: Take out the largest perfect square factor — √72 = 6√2
• Rationalizing: For 1/(√a+√b) → use conjugate (√a−√b)
• ⚡ Exam tip: When rationalizing 1/(√a + √b), the result is (√a − √b)/(a − b)

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Solving Equations with Indices

Example: Solve 2^(x+1) = 32 → 32 = 2⁵ → x+1 = 5 → x = 4

Example: Solve 3^(2x−1) = 81 → 81 = 3⁴ → 2x−1 = 4 → 2x = 5 → x = 2.5

Example: Solve 5^(x+2) − 5^x = 600 5^x(25 − 1) = 600 → 5^x × 24 = 600 → 5^x = 25 = 5² → x = 2

Nested Surds

Example: Simplify √(6 + 2√5) Let √(6 + 2√5) = √a + √b where a+b = 6 and ab = 5 t² − 6t + 5 = 0 → (t−1)(t−5) = 0 → t = 1 or 5 So √(6 + 2√5) = √5 + 1

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Harder Index Problems

Example: Simplify (2⁻² × 4^(1/2))^(1/3) = (2⁻² × 2¹)^(1/3) = (2⁻¹)^(1/3) = 2^(−1/3)

Logarithms and Indices Connection

If aⁿ = b, then logₐ(b) = n. The laws of logarithms are derived from the laws of indices:

• log(ab) = log a + log b
• log(a/b) = log a − log b
• log(aⁿ) = n log a

Example: If log 2 ≈ 0.3010, find log 8 = log(2³) = 3 log 2 = 3 × 0.3010 = 0.9030

Example: Simplify log 6 + log 5 − log 3 = log(6×5/3) = log 10 = 1

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