At A-Level, index laws and surds are rarely tested on their own. They appear inside bigger problems: writing an expression as a sum of powers of x ready to differentiate, solving index equations by matching bases, simplifying and expanding surds, and rationalising a denominator to give an exact answer in the form a + b√c.
Index laws and surd rules
- xᵃ × xᵇ = xᵃ⁺ᵇ and xᵃ ÷ xᵇ = xᵃ⁻ᵇ
- (xᵃ)ᵇ = xᵃᵇ and (ab)ⁿ = aⁿbⁿ
- x⁰ = 1, x⁻ⁿ = 1/xⁿ, x^(1/n) = ⁿ√x
- √(ab) = √a × √b and (√a)² = a
- (p + √q)(p − √q) = p² − q
Writing expressions as sums of powers of x
Before you can differentiate or integrate an expression, every term must be in the form axⁿ. Three moves get it there: turn roots into powers (√x = x^(1/2), 1/xᵏ = x^(−k)), split a fraction term by term over its denominator, and subtract the powers when dividing (xᵃ/xᵇ = xᵃ⁻ᵇ).
Splitting the fraction is the step students skip. Each term on the top must be divided by the whole denominator.
Index equations with a common base
To solve an equation such as 2ˣ × 8 = 4ˣ⁺¹, rewrite every number as a power of the same base: 8 = 2³ and 4 = 2². Combine using the index laws, then equate the powers on each side, since aᵐ = aⁿ means m = n. When the bases cannot be matched, logarithms are needed instead.
Simplifying and expanding surds
Simplify a surd by taking out the largest square factor: √50 = √25 × √2 = 5√2. Only like surds can be collected, such as 3√2 + 5√2 = 8√2. Expand brackets containing surds exactly as you would algebraic brackets, remembering that (√a)² = a.
Rationalising the denominator
To remove a surd from a denominator, multiply the top and bottom by the conjugate, which is the denominator with the sign between its terms changed. The denominator becomes rational because of the difference of two squares: (p + √q)(p − √q) = p² − q. Then expand the numerator and simplify to the required form, usually a + b√c.
Exact answers in context
Exam questions often place surds inside a geometry problem: a side length or area in surd form, where you expand, rationalise or use Pythagoras. Keep every step exact. A final answer such as 4 + √3 should never be rounded unless the question asks.
Worked examples
Example 1. Express (2x³ − x)/√x as a sum of powers of x.
- √x = x^(1/2), so the expression is (2x³ − x)/x^(1/2).
- Split: 2x³/x^(1/2) − x/x^(1/2).
- Subtract the powers: 2x^(5/2) − x^(1/2).
Answer: 2x^(5/2) − x^(1/2).
Marks: M1 for splitting and writing √x = x^(1/2), A1 for the answer.
Example 2. Solve 2ˣ × 8 = 4ˣ⁺¹.
- Write as powers of 2: 2ˣ × 2³ = (2²)ˣ⁺¹.
- 2ˣ⁺³ = 2²ˣ⁺².
- Equate the powers: x + 3 = 2x + 2.
Answer: x = 1.
Marks: M1 for the common base, M1 for x + 3 = 2x + 2, A1 for x = 1.
Example 3. Expand and simplify (2 + √3)(5 − 2√3).
- Expand all four products: 10 − 4√3 + 5√3 − 2(√3)².
- (√3)² = 3, so this is 10 − 4√3 + 5√3 − 6.
- Collect.
Answer: 4 + √3.
Marks: M1 for expanding all four terms, A1 for 4 + √3.
Example 4. Express (7 + √3)/(2 + √3) in the form a + b√3.
- Multiply top and bottom by the conjugate 2 − √3.
- Denominator: (2 + √3)(2 − √3) = 4 − 3 = 1.
- Numerator: (7 + √3)(2 − √3) = 14 − 7√3 + 2√3 − 3 = 11 − 5√3.
Answer: 11 − 5√3.
Marks: M1 for the conjugate, M1 for the denominator, A1 for 11 − 5√3.
Example 5. A right-angled triangle has shorter sides (1 + √3) cm and (3 − √3) cm. Find its exact area.
- Area = ½(1 + √3)(3 − √3).
- (1 + √3)(3 − √3) = 3 − √3 + 3√3 − 3 = 2√3.
- Halve it.
Answer: √3 cm².
Marks: M1 for setting up the area, A1 for √3.
Common mistakes
- Not splitting the fraction. Divide every term on the top by the denominator before simplifying.
- Adding unlike surds. √2 + √3 is not √5; only like surds combine.
- Multiplying by the wrong conjugate. For 2 + √3, multiply by 2 − √3, top and bottom.
- Forgetting (√a)² = a. −2√3 × √3 = −2 × 3 = −6.
- Mixing bases in an index equation. Write every term as a power of the same base before equating powers.
How indices and surds are examined at A-Level
Edexcel International A-Level opens Pure 1 (WMA11) with indices and surds, and Edexcel A-Level (9MA0) teaches them at the start of Year 1. Typical tasks:
- rewrite a fraction with roots as separate powers of x, often as part (a) of a calculus question
- rationalise a denominator to a stated form
- solve an index equation by using a common base
- find an exact length or area in surd form
The Pure 1 algebraic expressions exam questions cover these skills with full mark schemes. Use the algebraic expressions worksheet or the A-Level algebra revision engine for more practice.
These skills are used straight away in basic differentiation and integrating powers of x. For the GCSE version, see surds.
Frequently asked questions
How do you rationalise a denominator like 2 + √3?
Multiply the top and bottom by its conjugate, 2 − √3. The denominator becomes 4 − 3 = 1, a rational number.
What does x^(1/2) mean?
The square root of x. In general x^(1/n) is the nth root of x, and x^(m/n) is the nth root of x raised to the power m.
How do you solve 2ˣ = 8ˣ⁻¹?
Write 8 as 2³, so 2ˣ = 2³⁽ˣ⁻¹⁾. Equate the powers: x = 3x − 3, giving x = 3/2.