Exponents & Indices
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Lesson overview
The laws of indices including zero, negative and fractional indices, exponential equations, the laws of logarithms, and simplifying and rationalising surds.
Go to lesson quiz →Exam guidance · JAMB UTME Speed Lens
Pacing
~40 seconds per MCQ (180 questions in 120 minutes, no negative marking).
How JAMB marks this
Fast dimensional/sign elimination and standard formula shortcuts — skip multi-line algebra when substitution tests options faster.
Watch out for
UTME distractors routinely include the intermediate value (e.g., forgetting to convert cm³ to dm³ or °C to Kelvin).
Lesson content
Overview
Indices, logarithms and surds form one connected topic in the JAMB syllabus, and together they supply several questions every year. Indices describe repeated multiplication, logarithms undo indices, and surds are the roots that cannot be simplified to whole numbers. Candidates aiming for Medicine, Engineering and other competitive courses cannot afford to drop these marks. The rules are few; the skill is applying them in the right order without slips. This lesson states each rule, shows why it works, and drills it on UTME-style questions.
Key concept: indices, logarithms and surds
In aⁿ, a is the base and n is the index (power or exponent). For a positive whole number n, aⁿ means n factors of a multiplied together. Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ; a^(1/n) = ⁿ√a; a^(m/n) = (ⁿ√a)ᵐ. A logarithm is an index: log_a x = y means aʸ = x. So log₂8 = 3 because 2³ = 8. Logarithms to base 10 are written log x. Laws of logarithms: log(xy) = log x + log y; log(x/y) = log x − log y; log xⁿ = n log x; log_a a = 1; log_a 1 = 0. Change of base: log_a x = log_b x ÷ log_b a. A surd is an irrational root such as √2 or √5. Surds are simplified using √(ab) = √a × √b, and a fraction is rationalised by removing the surd from its denominator.
Using the rules
Fractional indices: take the root first, then the power, to keep the numbers small: 27^(2/3) = (∛27)² = 3² = 9, and 16^(3/4) = (⁴√16)³ = 2³ = 8. Algebraic terms follow the same laws, applied to the number and to each letter: (8x⁶)^(1/3) = ∛8 × x^(6/3) = 2x², and (a³b⁻²)² ÷ a⁴b⁻³ = a⁶b⁻⁴ ÷ a⁴b⁻³ = a²b⁻¹ = a²/b. Change of base lets you evaluate logarithms to awkward bases: log₄8 = log₂8 ÷ log₂4 = 3/2. Because 4^(3/2) = (√4)³ = 8, this agrees with the definition. Logarithms in calculations: the logarithm of a number in standard form A × 10ⁿ is n + log A, so log 2000 = 3 + log 2 = 3.3010. This is how four-figure tables turn multiplication into addition. Exponential equations: write both sides as powers of the same base, then equate the indices. If 3ˣ = 81, then 3ˣ = 3⁴ and x = 4. If the bases cannot be matched, take logarithms of both sides. Logarithms: combine everything into a single logarithm using the laws, then convert to index form. Given log 2 and log 3, any number made from 2s and 3s by multiplying or dividing can be found: log 6 = log 2 + log 3. Logarithm equations: the expression inside any logarithm must be positive, so always check the solutions and reject any that make a logarithm undefined. Simplifying surds: take out the largest square factor. √50 = √25 × √2 = 5√2 and √18 = 3√2, so √50 − √18 = 2√2. Only like surds can be added or subtracted. Rationalising: multiply the top and bottom by the conjugate of the denominator. The conjugate of √a − √b is √a + √b, and (√a − √b)(√a + √b) = a − b, which has no surd.
Formulae to know
aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ a⁰ = 1; a⁻ⁿ = 1/aⁿ; a^(m/n) = (ⁿ√a)ᵐ log_a x = y ⇔ aʸ = x log xy = log x + log y; log(x/y) = log x − log y; log xⁿ = n log x √(ab) = √a √b; (√a − √b)(√a + √b) = a − b
Worked example 1: fractional and negative indices
Evaluate 27^(2/3) × 16^(−3/4).
Active Worked Example · Predict Step 1
4 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Worked example 2: exponential equations
Solve (a) 2^(2x + 1) = 32 and (b) 9ˣ = 27^(x − 1).
Active Worked Example · Predict Step 1
5 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Worked example 3 (UTME standard): a logarithm equation
Solve log₂(x + 3) + log₂(x − 3) = 4.
Active Worked Example · Predict Step 1
5 stepsWrite down the formula, rule, or first move you would make before revealing Step 1.
Common traps
• Multiplying the base by the index: 3⁴ = 81, not 12. • Treating a negative index as a negative number: 2⁻³ = 1/8, not −8. • Writing log(x + y) = log x + log y. The law is log(xy) = log x + log y. • Adding unlike surds: √2 + √3 is not √5. • Forgetting to reject solutions that make a logarithm's argument zero or negative.
Summary
• Multiply: add indices; divide: subtract; power of a power: multiply. a⁰ = 1, a⁻ⁿ = 1/aⁿ, a^(m/n) = (ⁿ√a)ᵐ. • Solve exponential equations by writing both sides with the same base. • log_a x = y means aʸ = x; use the product, quotient and power laws, and check solutions. • Simplify surds by taking out square factors; rationalise with the conjugate.
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