M4 · Powers, roots and standard form | Edexcel GCSE Maths

Revise the key ideas

In 3⁴, 3 is the base and 4 is the index: 3 × 3 × 3 × 3 = 81. Squaring and cubing mean powers 2 and 3. A power is not base × index.
Square roots undo squaring: √49 = 7. The symbol √ gives the non-negative square root; the equation x² = 49 has two solutions, x = ±7. Cube roots can be negative: ∛(−8) = −2.
Recognise useful powers: 2⁵ = 32, 3³ = 27, 4³ = 64 and 5³ = 125. Know common squares and cubes so you can undo familiar powers without a calculator.
With the same base, multiply by adding indices and divide by subtracting: a^maⁿ = a^m+n; a^maⁿ = a^m−n for a ≠ 0.
A power of a power multiplies indices: (a^m)ⁿ = a^mn. Also (ab)ⁿ = aⁿbⁿ. You cannot apply these rules to sums: (a + b)² is not a² + b².
For a non-zero base, a⁰ = 1 and a⁻ⁿ = 1aⁿ. For example, 2⁻³ = 18. A negative index does not make the value negative.

Standard form is A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. 43 000 = 4.3 × 10⁴; 0.00072 = 7.2 × 10⁻⁴.
Positive powers of 10 describe large numbers; negative powers describe small positive numbers. 0.003 is 3 × 10⁻³, not −3000.
Worked example: (3 × 10⁵)(4 × 10⁻²) = 12 × 10³ = 1.2 × 10⁴. Multiply coefficients, add indices, then adjust the coefficient to the required range.
Worked example: 6 × 10⁷2 × 10³ = 3 × 10⁴. Divide coefficients and subtract indices.
For addition/subtraction, rewrite with the same power of 10: 3.2 × 10⁴ + 6 × 10³ = (3.2 + 0.6) × 10⁴ = 3.8 × 10⁴.
On a calculator, use its standard-form entry key and brackets around complete numbers when dividing. Check whether the displayed exponent is positive or negative.

Estimate roots using nearby known powers: since 8² < 70 < 9², √70 lies between 8 and 9. Trial values can refine an estimate without implying the root is rational.
For a positive base, a¹ⁿ means the nth root of a, and a^mn means raise the nth root to power m. Thus 27²³ = (∛27)² = 9.
Worked example: 16⁻³⁴ = 1(⁴√16)³ = 12³ = 18. Handle the negative sign in the index by taking a reciprocal.
Fractional and negative indices obey the same index laws within their real-number domains. For example, x¹² × x³² = x² for x > 0.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.