1.8 Squares, cubes and roots

Squares and cubes

Squaring a number means multiplying it by itself. We write 7^2 (said '7 squared'), which means 7 \times 7 = 49. It is called a square number because it is the area of a square with sides of that length. A cube number such as 4^3 = 4 \times 4 \times 4 = 64 is the volume of a cube.

You are expected to recall the square numbers up to 15 \times 15 without a calculator.

The remaining squares are 11^2 = 121, 12^2 = 144, 13^2 = 169, 14^2 = 196 and 15^2 = 225. Notice that 64 is both a square number and a cube number: 8^2 = 64 and 4^3 = 64.

Other powers

A power shows repeated multiplication. 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32. You should recognise the powers of 2, 3, 4 and 5.

Powers of 10. 10^2 = 100, 10^3 = 1000, 10^4 = 10\,000, and so on. You should know that 1000 = 10^3 and 1 million = 10^6. The power tells you how many zeros follow the 1.

Roots

A root is the inverse of a power. The square root of 49 is written \sqrt{49} = 7 because 7^2 = 49. The cube root of 64 is written \sqrt[3]{64} = 4 because 4^3 = 64. Higher roots work in the same way: \sqrt[4]{81} = 3 because 3^4 = 81.

Square roots have two values. Because (-7) \times (-7) = 49 as well, the equation x^2 = 49 has two solutions: x = 7 or x = -7, written x = \pm 7. The symbol \sqrt{\ } on its own means the positive root.

Worked example. Work out \sqrt{144} + \sqrt[3]{27}.

\sqrt{144} = 12 and \sqrt[3]{27} = 3, so the answer is 12 + 3 = 15.

Worked example. Work out 5^3 - 4^2.

5^3 = 125 and 4^2 = 16, so 125 - 16 = 109.

Worked example. A square has area 169 cm². What is the length of its side?

The side is \sqrt{169} = 13 cm. A cube has volume 216 cm³; its edge is \sqrt[3]{216} = 6 cm.

Estimating roots

If a number is not a perfect square, its root lies between two integers. Find the nearest square numbers on each side.

Worked example. Estimate \sqrt{50}.

49 < 50 < 64, so 7 < \sqrt{50} < 8. Since 50 is much closer to 49 than to 64, \sqrt{50} is just above 7: about 7.1. A calculator gives 7.07 (to 2 d.p.).

Worked example. Between which two integers does \sqrt[3]{30} lie?

27 < 30 < 64, so 3 < \sqrt[3]{30} < 4. It is close to 3, about 3.1.

Using a calculator

Use the x^2, x^3 and x^{\square} keys for powers and the \sqrt{\ }, \sqrt[3]{\ } and \sqrt[\square]{\ } keys for roots. Check that you can reproduce these answers: 2^{10} = 1024, \sqrt{2025} = 45 and \sqrt[3]{1728} = 12.

Common mistakes

• Writing 4^3 = 4 \times 3 = 12. The power means 4 \times 4 \times 4 = 64.
• Thinking the square root of 100 is 50. Halving is not the same as taking a root.
• Forgetting that (-5)^2 = 25 is positive, because a negative times a negative is positive.
• Mixing up 2^3 = 8 and 3^2 = 9.

Exam-style example

Question. (a) Write down the value of 3^4. (b) Write down a number that is both a square number and a cube number, and is greater than 1. (c) Show that \sqrt{90} is between 9 and 10.

Answer. (a) 3^4 = 3 \times 3 \times 3 \times 3 = 81. (b) 64, because 8^2 = 64 and 4^3 = 64. (c) 9^2 = 81 and 10^2 = 100, and 81 < 90 < 100, so 9 < \sqrt{90} < 10.

Why are they called squares and cubes?

A square number gets its name from geometry. If you arrange counters in a square grid, the number of counters is a square number: a 3 by 3 grid has 9 counters and a 4 by 4 grid has 16. The side length of the square is the number you squared. In the same way, a cube number counts the small unit cubes in a larger cube: a cube with edges of 3 units is made of 27 unit cubes, because 3 \times 3 \times 3 = 27. This is why finding a square root or cube root is the same as finding the side length of a square with a given area or the edge length of a cube with a given volume.

Patterns in the square numbers

Look at the differences between neighbouring square numbers: 1, 4, 9, 16, 25, 36 have gaps of 3, 5, 7, 9, 11. These gaps are the odd numbers, so you can build each square number from the one before it. For example 8^2 = 7^2 + 15 = 49 + 15 = 64. This also shows that adding the first n odd numbers always gives n^2: 1 + 3 + 5 + 7 = 16 = 4^2.

The last digit of a square number can only be 0, 1, 4, 5, 6 or 9. So a number such as 2 347 cannot be a perfect square because it ends in 7. This is a quick way to rule out answers in a question about square numbers.

A step-by-step method for estimating roots

1. Write down the square numbers (or cube numbers) either side of the number you are given.
2. State the two integers that the root lies between.
3. Decide which of the two the number is closer to, and give an estimate.
4. Check by squaring your estimate: 7.1^2 = 50.41, which is close to 50.

Where these facts are used

Square numbers and square roots are used throughout GCSE mathematics: finding the side of a square from its area, Pythagoras' theorem, the quadratic formula, the equation of a circle and the laws of indices. Cube numbers appear in volume calculations and in questions about the volume of a cube. If you know the squares up to 15 and the cubes up to 10 instantly, many of these questions become much quicker.

Summary

• Know the squares up to 15^2, the cubes up to 10^3, and the powers of 2, 3, 4 and 5 in the tables.
• 1000 = 10^3 and 1 million = 10^6.
• Roots are the inverse of powers: \sqrt{49} = 7, \sqrt[3]{64} = 4.
• The equation x^2 = k has two solutions, x = \pm \sqrt{k}.
• To estimate a root, find the two perfect powers on either side.