Powers, Roots & Standard FormEdexcel GCSE Maths: Revision notes
Section 1
What are powers and roots?
A power (or index/exponent) shows how many times a number is multiplied by itself. A root is the inverse of a power.
- Square: 5² = 25; square root: √25 = 5
- Cube: 3³ = 27; cube root: ∛27 = 3
Memorise powers of 2, 3, 4 and 5 (e.g. 2² to 2⁵, squares up to 15², cubes up to 5³) to estimate powers and roots of any positive number quickly, even without a calculator.
To estimate a root, find the two consecutive integers whose powers the number falls between, e.g. √50 is between 7 (49) and 8 (64), so √50 ≈ 7.1.
Section 2
How do we calculate with integer and fractional indices?
Fractional indices represent roots: a^(1/n) = ⁿ√a. Combined indices like a^(m/n) mean take the nth root, then raise to the mth power (or vice versa).
- 8^(1/3) = ∛8 = 2
- 8^(2/3) = (∛8)² = 2² = 4
Negative indices represent reciprocals: a⁻ⁿ = 1/aⁿ.
- 2⁻³ = 1/2³ = 1/8
Section 3
What are the index laws?
| Law | Rule | Example |
|---|---|---|
| Multiplying | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2² = 2⁵ |
| Dividing | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 2⁵ ÷ 2² = 2³ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ |
| Zero index | a⁰ = 1 | 5⁰ = 1 |
| Negative index | a⁻ⁿ = 1/aⁿ | 3⁻² = 1/9 |
| Fractional index | a^(1/n) = ⁿ√a | 27^(1/3) = 3 |
These laws only apply directly when the base numbers are the same.
A common error is adding indices when multiplying different bases (e.g. 2³ × 3² ≠ 6⁵) — the index laws for multiplying/dividing only work when the base is identical.
Section 4
What is standard form and how do we calculate with it?
Standard form writes very large or very small numbers as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
- 5,600,000 = 5.6 × 10⁶
- 0.00042 = 4.2 × 10⁻⁴
To convert standard form to an ordinary number, move the decimal point n places right (positive n) or left (negative n).
To multiply/divide numbers in standard form: multiply/divide the A values, and add/subtract the powers of 10, then adjust so A is back between 1 and 10.
To add/subtract numbers in standard form: convert to the same power of 10 first, then add/subtract the A values.
(3 × 10⁵) × (2 × 10³) = 6 × 10⁸. (8 × 10⁶) ÷ (4 × 10²) = 2 × 10⁴
Must Know
- aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ (same base only)
- a⁰ = 1 for any non-zero a
- a⁻ⁿ = 1/aⁿ (negative index gives a reciprocal)
- a^(1/n) = ⁿ√a (fractional index gives a root)
- Standard form: A × 10ⁿ where 1 ≤ A < 10 and n is an integer
- To add/subtract in standard form, match the powers of 10 first
That's the notes covered.
Carry on to the next subtopic.