Sheet № 04 · Foundation + Higher · AQA · Edexcel · OCR
Powers and Roots
Powers and roots sit at the heart of GCSE Maths. Every student — Foundation and Higher — needs to recognise square numbers, cube numbers, and their roots, and apply them in calculations ranging from area problems to Pythagoras' theorem. Higher tier students also meet fractional and negative powers, but this page focuses on building rock-s
§Key definitions
Question:
Work out the value of 4³ − √49.
Q1 (Foundation):
Work out the value of 6² + ∛64.
Q2 (Foundation):
Put these in order from smallest to largest: √36, 2³, 3², √100.
Q3 (Higher):
Without a calculator, estimate √(150) to one decimal place. You must show your working.
§Formulas to memorise
a^n means a multiplied by itself n times
√a × √a = a
∛a × ∛a × ∛a = a
√(a × b) = √a × √b
Square number: — A number multiplied by itself, e.g. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
Cube number: — A number multiplied by itself three times, e.g. 1, 8, 27, 64, 125.
Square root (√): — The value that, when squared, gives the original number.
Cube root (∛): — The value that, when cubed, gives the original number.
Write out the multiplication in full. For example, 2⁴ = 2 × 2 × 2 × 2.
Multiply step by step: 2 × 2 = 4, 4 × 2 = 8, 8 × 2 = 16.
State the result: 2⁴ = 16.
Worked example
Work out the value of 4³ − √49.
Working:
⚠ Common mistakes
- ✗Confusing squaring with doubling. 5² = 25, not 10. Squaring means multiplying the number by itself, not by 2.
- ✗Forgetting that every positive number has two square roots. √25 = 5, but −5 is also a square root of 25 because (−5)² = 25. Questions usually want the positive root unless stated otherwise.
- ✗Thinking that (−3)² and −3² are the same. (−3)² = 9 (the negative is inside the brackets and gets squared), but −3² = −9 (the square applies only to the 3).
- ✗Not recognising that √0 = 0 and 0² = 0. These come up more often than you might expect.
- ✗Mixing up square roots and cube roots. The cube root of 27 is 3 (because 3³ = 27), not 9.
✦ Exam tips
- →Memorise square numbers up to 15² = 225 and cube numbers up to 5³ = 125. These appear regularly in non-calculator papers and speed up many other topics like Pythagoras and area.
- →When estimating roots on non-calculator papers, show your reasoning. Write down the perfect squares or cubes you are using as bounds — this earns method marks.
- →Check whether a question says "positive square root" or just "square root". In GCSE, √ usually means the positive root, but read the question carefully.
- →Connect this topic to others. Powers and roots appear in index laws, surds, and area/volume calculations. See the grade boundaries guide to understand how different topics contribute to your overall grade.
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