1.3 Indices and standard form
Indices and standard form
Everything on indices and standard form that can come up in the WJEC GCSE Mathematics and Numeracy, in Unit 2 and Unit 3. Each skill below has the method, the mistake that costs the most marks, a worked example, and free practice marked the way WJEC marks it.
Practise indices and standard form free
A worked example
Index laws · [1 mark]
Write 6² × 6⁷ as a single power of 6.
- B16⁹
The exam skills (5)
Index laws
- Multiplying: add the powers. Dividing: subtract them.
- Power of a power: multiply the powers.
- Keep the same base number.
Watch out: Multiplying the base numbers: 2² × 2² is 2⁴, not 4⁴.
Standard form
- Standard form is A × 10ⁿ, with 1 ≤ A < 10.
- Move the decimal point until you have a number between 1 and 10.
- Count the jumps: big numbers get a positive power, small numbers negative.
Watch out: Writing 11 × 10⁻⁶. The first number must be less than 10.
Zero and negative powers
- Anything to the power 0 is 1.
- A negative power means 1 over: 2⁻³ = 1/2³.
- Then work out the power.
Watch out: Thinking 2⁻³ is negative. It is a small positive fraction: 1/8.
Fractional indices
- The bottom of the fraction is a root: x^(1/2) = √x, x^(1/3) = ∛x.
- The top of the fraction is a power.
- Do the root first, then the power, so the numbers stay small.
Watch out: Dividing by the bottom number. 9^(1/2) is √9 = 3, not 9 ÷ 2.
Solving with indices
- Write both sides as powers of the same number, e.g. 8 = 2³.
- If the bases match, the powers must be equal.
- Solve for x.
Watch out: Dividing 8 by 2. Powers are repeated multiplying, not multiplying once.
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