At a glance
- Cambridge 0580
- Core C1.7, C1.8, C2.4; Extended E1.7, E1.18, E2.4
- Edexcel 4MA1
- Foundation 1.4C, 2.1C, D; Higher 1.4A to C
- Negative power
- a^-n = 1/a^n
- Fractional power
- a^(m/n) = (nth root of a)^m
What each tier expects
| Board and tier | Reference | What can be asked |
|---|---|---|
| Cambridge 0580 Core | C1.7, C1.8, C2.4 | Positive, zero and negative integer indices; index laws with numbers and letters, e.g. (5x^3)^2; standard form; solve 2^x = 32 |
| Cambridge 0580 Extended | E1.7, E1.18, E2.4 | Fractional indices such as 8^(2/3); surds, simplifying and rationalising; equations such as 5^(x + 1) = 25^x |
| Edexcel 4MA1 Foundation | 1.4C, 1.9, 2.1C, 2.1D | Index laws with positive, negative and zero integer powers; standard form |
| Edexcel 4MA1 Higher | 1.4A to 1.4C, 2.1A | Surds, rationalising denominators, fractional and negative powers, e.g. 625^(-1/2) |
The key ideas
The index laws apply only to the same base. x^3 × x^4 = x^7, x^7 ÷ x^2 = x^5 and (x^3)^4 = x^12. Numbers in front are dealt with separately: 6x^7 × 5x^-5 = 30x^2. When a bracket is raised to a power, every factor inside is raised to it: (5x^3)^2 = 25x^6, not 5x^6.
Negative and fractional powers are definitions that keep the laws consistent. a^0 = 1 because a^n ÷ a^n = 1. a^-2 = 1/a^2. a^(1/2) = sqrt(a) because a^(1/2) × a^(1/2) = a. For a^(m/n), take the root first, which keeps numbers small: 8^(2/3) = (cube root of 8)^2 = 2^2 = 4.
To solve an equation with unknown powers, write both sides with the same base and then equate the powers. 25 = 5^2, so 5^(x + 1) = 25^x becomes 5^(x + 1) = 5^(2x), and x + 1 = 2x.
Surds: simplify by taking out the largest square factor, so sqrt(200) = sqrt(100 × 2) = 10 sqrt(2). Collect like surds as you would like terms. To rationalise a/sqrt(b), multiply top and bottom by sqrt(b). To rationalise 1/(sqrt(3) - 1), multiply top and bottom by sqrt(3) + 1, because (sqrt(3) - 1)(sqrt(3) + 1) = 3 - 1 = 2.
Worked example 1: evaluate and simplify without a calculator
- 27^(-2/3): take the cube root, 3; square it, 9; the negative power means reciprocal. Answer: 1/9.
- (1/25)^(-1/2): the negative power flips the fraction to 25^(1/2) = 5. Answer: 5.
- 12a^5 ÷ 3a^-2 = (12 ÷ 3) × a^(5 - (-2)) = 4a^7.
- (3 × 10^4) × (4 × 10^-7) = 12 × 10^-3 = 1.2 × 10^-2 in standard form.
Worked example 2: solve equations with indices
- Solve 5^(x + 1) = 25^x. Write 25 as 5^2: 5^(x + 1) = 5^(2x). Equate powers: x + 1 = 2x, so x = 1. Check: 5^2 = 25 and 25^1 = 25. Correct.
- Solve 32^x = 2. Write 32 as 2^5: 2^(5x) = 2^1, so 5x = 1 and x = 0.2. Check: 32^0.2 is the fifth root of 32, which is 2. Correct.
- Solve 2^x = 1/8. 1/8 = 2^-3, so x = -3.
Worked example 3: surds
- Simplify sqrt(200) - sqrt(32): 10 sqrt(2) - 4 sqrt(2) = 6 sqrt(2).
- Expand (3 + 5 sqrt(2))^2: 9 + 2 × 3 × 5 sqrt(2) + 25 × 2 = 9 + 30 sqrt(2) + 50 = 59 + 30 sqrt(2).
- Rationalise 10/sqrt(5): multiply top and bottom by sqrt(5) to get 10 sqrt(5) ÷ 5 = 2 sqrt(5).
- Rationalise 1/(sqrt(3) - 1): multiply top and bottom by sqrt(3) + 1. The bottom becomes 3 - 1 = 2, so the answer is (1 + sqrt(3))/2. Check with a calculator: both give 1.366...
Common mistakes that cost marks
- Multiplying the powers when multiplying terms: x^3 × x^4 is x^7, not x^12.
- Applying the power to only part of a bracket: (2x)^3 is 8x^3, not 2x^3.
- Treating a negative power as a negative number: 2^-3 is 1/8, not -8.
- Writing sqrt(a + b) = sqrt(a) + sqrt(b). sqrt(9 + 16) is 5, not 7.
- Not taking out the largest square factor: sqrt(72) = 6 sqrt(2), not 2 sqrt(18).
- Multiplying only the denominator when rationalising.
Exam technique and how a tutor helps
Indices and surds are classic non-calculator questions on Cambridge Papers 1 and 2. Show the intermediate step, such as '(cube root of 27)^2 = 9', because the method mark is often given for it. Learn the squares to 15^2 and cubes of 1 to 5 and 10, which Cambridge expects you to recall, and the powers of 2 to 2^10.
On Edexcel 4MA1 every paper allows a calculator, but surd questions still say 'show your working' or ask for an exact answer, so the algebra must be visible.
This topic rewards fluency, and fluency comes from short, frequent practice. In one-to-one lessons a tutor can run quick-fire rounds on the shared whiteboard, watch for the specific law that is being misapplied, and build up from integer powers to fractional powers to surds in a sensible order rather than all at once.
Self-check: can you do these?
- Find the value of 2^-3 × 2^4. (Answer: 2)
- Find the value of 125^(2/3). (Answer: 25)
- Simplify (5x^3)^2. (Answer: 25x^6)
- Rationalise 6/sqrt(3). (Answer: 2 sqrt(3))
- Expand and simplify (sqrt(7) + 1)(sqrt(7) - 1). (Answer: 6)
Common questions
What does a fractional index mean?
The denominator is a root and the numerator is a power. 8^(2/3) means the cube root of 8, squared, which is 4. Fractional indices are Cambridge Extended and Edexcel Higher content.
Are surds on the Core or Foundation tier?
No. Surds are Cambridge 0580 Extended (E1.18) and Edexcel 4MA1 Higher (1.4A, 1.4B). Cambridge Extended can also ask for quadratic solutions in surd form.
Do I need logarithms for index equations?
No. Cambridge says knowledge of logarithms is not required; index equations at IGCSE are solved by writing both sides with the same base.
Why do we rationalise the denominator?
It gives a standard simplest form that is easier to compare and calculate with. Questions will say 'rationalise the denominator' or ask for the simplest form.
How do LiveTutor lessons work?
Lessons are one to one, online and 60 minutes, with a tutor who teaches your child's board, at $15 each. The first lesson is a free trial.
Sources
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