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-solid understanding of what powers and roots mean, how to evaluate them, and where they appear in exams. If you want a full revision plan, our how to revise GCSE Maths guide is a great starting point.

What Are Powers and Roots?

A power (or index) tells you how many times to multiply a number by itself. In the expression 5³, the base is 5 and the power is 3, meaning 5 × 5 × 5 = 125. We say "five cubed" or "five to the power of three".

A root is the reverse operation. The square root of 25 is 5 because 5² = 25. The cube root of 8 is 2 because 2³ = 8.

Key Definitions

• 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.

Key Formulas

Important Values to Memorise

You should know the first 15 square numbers (1 to 225) and the first 5 cube numbers (1, 8, 27, 64, 125) by heart. Recognising these instantly saves valuable time in exams.

Step-by-Step Method

Evaluating Powers

1. Write out the multiplication in full. For example, 2⁴ = 2 × 2 × 2 × 2.
2. Multiply step by step: 2 × 2 = 4, 4 × 2 = 8, 8 × 2 = 16.
3. State the result: 2⁴ = 16.

Finding Square Roots Without a Calculator

1. Ask yourself: "What number times itself gives this value?"
2. If the number is not a perfect square, estimate by finding the two consecutive square numbers it falls between.
3. Narrow down using trial and improvement. For example, √50 is between 7 (49) and 8 (64), closer to 7, so approximately 7.07.

Finding Cube Roots

1. Ask: "What number cubed gives this value?"
2. Use known cube numbers to identify or estimate the answer.

Combining Powers and Roots

When a calculation includes both, evaluate powers and roots before addition and subtraction (following BIDMAS). For example, 3² + √16 = 9 + 4 = 13.

Worked Example 1 — Foundation Level

Question: Work out the value of 4³ − √49.

Working:

Step 1 — Evaluate 4³: 4 × 4 × 4 = 64.

Step 2 — Evaluate √49: 7 × 7 = 49, so √49 = 7.

Step 3 — Subtract: 64 − 7 = 57.

Answer: 57

Worked Example 2 — Higher Level

Question: Estimate the value of √(90). Give your answer to one decimal place.

Working:

Step 1 — Identify the nearest perfect squares: 9² = 81 and 10² = 100. So √90 is between 9 and 10.

Step 2 — Since 90 is closer to 81 than to 100, √90 is closer to 9 than to 10.

Step 3 — Try 9.5: 9.5² = 90.25. That is just above 90.

Step 4 — Try 9.4: 9.4² = 88.36. Too low.

Step 5 — Try 9.49: 9.49² = 90.0601. Very close.

Step 6 — Since 9.49² ≈ 90, √90 ≈ 9.5 to one decimal place.

Answer: 9.5

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.

Practice Questions

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.

Build your confidence with powers and roots using adaptive AI practice. Start for free and get instant feedback on every question.

Related Topics

• Indices and Index Laws
• Surds
• Order of Operations (BIDMAS)
• Standard Form

Summary

• A power tells you how many times to multiply a base number by itself.
• Square numbers result from multiplying a number by itself; cube numbers from multiplying three times.
• A square root reverses squaring; a cube root reverses cubing.
• Memorise squares up to 15² and cubes up to 5³ for speed in exams.
• Every positive number has two square roots (positive and negative), but √ notation typically means the positive one.
• Use known perfect squares or cubes to estimate non-perfect roots.
• Powers and roots are evaluated before addition and subtraction in BIDMAS.