A square number is a number multiplied by itself, written n², and a cube number is n × n × n, written n³. A square root reverses squaring: √49 = 7, because 7² = 49. A cube root reverses cubing: ∛125 = 5. To estimate a root that is not a whole number, find the square or cube numbers on either side.
Powers and roots
- Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
- Cubes: 1, 8, 27, 64, 125, 1000
- √ reverses squaring; ∛ reverses cubing
- A positive number has two square roots, one positive and one negative
Square and cube numbers
A square number is a whole number multiplied by itself: 6² = 36. A cube number is a whole number multiplied by itself three times: 4³ = 64. Learn the squares up to 15² and the cubes 1, 8, 27, 64, 125 and 1000; they appear constantly on non-calculator papers.
Square roots and cube roots
A square root asks which number, multiplied by itself, gives this number. √81 = 9 because 9 × 9 = 81. A cube root asks which number, used three times, gives this number: ∛64 = 4.
Every positive number has two square roots, one positive and one negative, since (−9)² = 81 as well. The √ symbol means the positive root. A negative number has a negative cube root: ∛(−8) = −2.
Estimating roots
Most numbers do not have a whole-number root. To estimate √50, find the square numbers either side: 49 = 7² and 64 = 8². So √50 is between 7 and 8, and very close to 7. The same method works for cube roots, using cube numbers.
Higher powers
The index tells you how many times to multiply the base by itself: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Work step by step to avoid errors, and remember that 2⁵ is not 2 × 5.
Using a calculator
Calculators have keys for squares, cubes, square roots and general powers. Use brackets for negative numbers, so that (−3)² gives 9 rather than −9. For roots of expressions, put the whole expression under the root, such as √(16 + 9) = √25 = 5.
Worked examples
Example 1. Work out 4² + 3³.
- 4² = 16.
- 3³ = 27.
- 16 + 27.
Answer: 43.
Example 2. Work out √81 and ∛125.
- 9 × 9 = 81, so √81 = 9.
- 5 × 5 × 5 = 125, so ∛125 = 5.
Answer: 9 and 5.
Example 3. Between which two whole numbers does √50 lie?
- 49 = 7² and 64 = 8².
- 50 lies between 49 and 64.
Answer: Between 7 and 8.
Example 4. Work out 2⁵ − √36.
- 2⁵ = 32.
- √36 = 6.
- 32 − 6.
Answer: 26.
Example 5. Find both square roots of 196.
- 14 × 14 = 196.
- (−14) × (−14) = 196 as well.
Answer: 14 and −14.
Common mistakes
- Treating a power as multiplication. 3³ = 27, not 9.
- Forgetting the negative square root. x² = 196 has solutions 14 and −14.
- Estimating to the wrong square. Find the square numbers on both sides before deciding.
- Missing brackets on a calculator. Type (−3)² to get 9.
- Square-rooting each term separately. √(16 + 9) = 5, not 4 + 3.
How powers and roots are examined
Powers and roots are Foundation tier content in Edexcel GCSE (1MA1) and the International GCSE (4MA1), often on the non-calculator paper. You may need to:
- work out squares, cubes, square roots and cube roots
- combine powers and roots in one calculation
- state which two whole numbers a root lies between
- identify square or cube numbers in a list
Practise with the powers and roots worksheet, which includes answers, or the number revision engine.
Powers lead into index laws, and roots into surds and Pythagoras.
Frequently asked questions
What is a square number?
A number made by multiplying a whole number by itself, such as 49 = 7 × 7.
What is a cube root?
The number that, multiplied by itself three times, gives the original. ∛27 = 3.
How do you estimate a square root?
Find the square numbers on either side. √30 is between 5 and 6, because 25 < 30 < 36.