Powers and Roots: Edexcel GCSE Foundation maths revision notes

What powers and roots are

A power is a short way of writing a number multiplied by itself. 6 \times 6 is written 6^2, and 6^2 = 36

A root takes you back again. The square root of 36 is 6, because 6 \times 6 = 36. Powers and roots are opposites, just as adding and subtracting are.

In the exam you are asked to work out a power such as 5^3, find a square root or a cube root, or write down a square number or a cube number from a list, usually for 1 mark each. Some questions show someone's claim, such as a wrong power, and ask whether they are correct, with a reason. Harder questions use a root in a problem, such as finding the side of a cube from its volume, for 3 or 4 marks.

Squares, cubes and other powers

In 5^3 the 5 is the number being multiplied, and the small 3 is the index, or power. The index tells you how many 5s to multiply together. It is read as 5 to the power 3.

5^3 = 5 \times 5 \times 5 = 125

A number to the power 2 is squared: 7^2 = 7 \times 7 = 49. A number to the power 3 is cubed: 4^3 = 4 \times 4 \times 4 = 64. Never multiply by the index: 7^2 is not 7 \times 2

For a higher power, multiply one step at a time and keep a running total. For 4^4: 4 \times 4 = 16, then 16 \times 4 = 64, then 64 \times 4 = 256

Worked example 1: Checking a claim about a power

Mia says that 2^5 = 10. Is Mia correct? Give a reason for your answer.

2^5 means five 2s multiplied together: 2 \times 2 \times 2 \times 2 \times 2
Multiply one step at a time: 2 \times 2 = 4, 4 \times 2 = 8, 8 \times 2 = 16
Your turn. Work out the last multiplication, 16 \times 2

Your answer

16 \times 2 = 32, so 2^5 = 32
Mia has worked out 2 \times 5 instead of multiplying five 2s together.
Answer: No, because 2^5 = 32, not 10

Powers of 10, negative numbers and calculators

Powers of 10 are quick: the index tells you how many zeros to write after the 1. 10^2 = 100, 10^3 = 1000, and 10^6 = 1\,000\,000, which is one million.

A negative number squared is positive, because a negative times a negative is positive: (-5)^2 = (-5) \times (-5) = 25. A negative number cubed stays negative: (-2)^3 = (-2) \times (-2) \times (-2) = -8

On a calculator paper, use the x^2 key to square, the x^3 key to cube, and the power key, marked x^y on many calculators, for other powers. For example, 1.2^3 = 1.728. There are keys for square roots and cube roots too.

Square numbers and cube numbers

Square numbers are what you get when you square a whole number. Learn the squares from 1^2 to 15^2: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

Cube numbers are what you get when you cube a whole number. Learn 1, 8, 27, 64, 125 and 1000, which are 1^3, 2^3, 3^3, 4^3, 5^3 and 10^3

They get their names from shapes. 16 small squares make a 4 by 4 square, and 27 small cubes make a 3 by 3 by 3 cube.

Don't forget 1. It is a square number, because 1 \times 1 = 1, and a cube number, because 1 \times 1 \times 1 = 1

Worked example 2: Square numbers and cube numbers in a list

Here is a list of numbers: 8, 20, 36, 45, 64, 81, 100, 125. Write down the number in the list that is both a square number and a cube number.

The square numbers in the list are 36 = 6^2, 64 = 8^2, 81 = 9^2 and 100 = 10^2
20 and 45 are not square numbers: 20 is between 16 and 25, and 45 is between 36 and 49
Your turn. Write down all the cube numbers in the list. Separate them with commas.

Your answer

Give every value, separated by commas

The cube numbers are 8 = 2^3, 64 = 4^3 and 125 = 5^3
Only 64 is in both lists: 8 \times 8 = 64 and 4 \times 4 \times 4 = 64
Answer: 64

Square roots

Finding a square root undoes squaring. 9^2 = 81, so the square root of 81 is 9. It is written with a root sign: \sqrt{81} = 9. The root sign always means the positive root, even though (-9)^2 = 81 too.

The area of a square is its side length squared. So if you know the area of a square, its side length is the square root of the area. A square with an area of 25\text{ cm}^2 has sides of \sqrt{25} = 5 cm.

Check a square root by squaring your answer. \sqrt{64} = 8, because 8 \times 8 = 64. It is not 32, which is half of 64, and it is not 4, which is the cube root of 64

Worked example 3: Square roots of a decimal and a fraction

Work out (a) \sqrt{0.64} (b) \sqrt{\frac{16}{49}}

(a) Start with the digits, ignoring the decimal point: \sqrt{64} = 8
Your turn. (a) Which decimal, multiplied by itself, gives 0.64?

Your answer

0.8 \times 0.8 = 0.64, so \sqrt{0.64} = 0.8
It is not 0.08, because 0.08 \times 0.08 = 0.0064
(b) Take the square root of the top and of the bottom: \sqrt{16} = 4 and \sqrt{49} = 7
So \sqrt{\frac{16}{49}} = \frac{4}{7}. Check: \frac{4}{7} \times \frac{4}{7} = \frac{16}{49}
Answer: (a) 0.8 (b) \frac{4}{7}

Estimating square roots

Most numbers are not square numbers, so their square roots are not whole numbers. You can still say which two whole numbers a square root lies between.

Find the square numbers just below and just above. 30 is between 25 and 36, so \sqrt{30} is between \sqrt{25} = 5 and \sqrt{36} = 6. This is written 5 < \sqrt{30} < 6

To estimate more closely, see which square number is nearer. 30 is 5 above 25 and 6 below 36, so \sqrt{30} is close to halfway between 5 and 6. A calculator gives \sqrt{30} = 5.477\ldots

Worked example 4: Estimating a square root

Write down the two whole numbers that \sqrt{140} lies between. Is \sqrt{140} nearer to the smaller one or the larger one?

Your turn. What is the largest square number less than 140?

Your answer

11^2 = 121, which is less than 140
The next square number is 12^2 = 144, which is more than 140
121 < 140 < 144, so 11 < \sqrt{140} < 12
140 is 19 more than 121 but only 4 less than 144, so \sqrt{140} is much nearer to 12
Answer: between 11 and 12, nearer to 12

Cube roots

Finding a cube root undoes cubing. 4^3 = 64, so the cube root of 64 is 4. It is written \sqrt[3]{64} = 4

Check a cube root by cubing your answer. \sqrt[3]{1000} = 10, because 10 \times 10 \times 10 = 1000. Dividing by 3 does not find a cube root: 1000 \div 3 is about 333, far too big.

Decimals work in the same way: \sqrt[3]{0.027} = 0.3, because 0.3 \times 0.3 \times 0.3 = 0.027

The volume of a cube is its side length cubed. So if you know the volume of a cube, its side length is the cube root of the volume.

Worked example 5: Using a cube root to find a side

A cube has a volume of 216\text{ cm}^3. Work out the total length of all the edges of the cube.

The volume is the side length cubed, so the side length is \sqrt[3]{216} cm.
Your turn. Work out \sqrt[3]{216}

Your answer

6 \times 6 \times 6 = 216, so each edge is 6 cm long.
A cube has 12 edges, all the same length: 4 round the top, 4 round the bottom and 4 going up.
Total length = 12 \times 6 = 72\text{ cm}
Don't divide 216 by 3, 6 or 12: find the side length first, with a cube root.
Answer: 72\text{ cm}

Worked example 6: Finding the volume of a cube from its faces

Ravi paints all six faces of a wooden cube. The total area he paints is 486\text{ cm}^2. Work out the volume of the cube.

A cube has 6 faces, all the same size. Area of one face = 486 \div 6 = 81\text{ cm}^2
Your turn. Each face is a square with an area of 81\text{ cm}^2. Work out the length of one edge of the cube, in cm.

Your answer

Edge length = \sqrt{81} = 9 cm
Volume = 9^3 = 9 \times 9 \times 9
9 \times 9 = 81, then 81 \times 9 = 729
Answer: 729\text{ cm}^3

Common mistakes

Multiplying by the index. 5^3 means 5 \times 5 \times 5 = 125, not 5 \times 3 = 15
Halving to find a square root. \sqrt{64} = 8, not 64 \div 2 = 32, and not the cube root, 4. Check by squaring your answer.
Dividing by 3 to find a cube root. \sqrt[3]{125} = 5, because 5 \times 5 \times 5 = 125. Check by cubing your answer.
Writing a power instead of its value. When a question asks for a square number, write 16, not 4^2, which is easily misread as 42
Forgetting that 1 is a square number. 1 \times 1 = 1, so 1 is a square number and a cube number.
Dividing the volume of a cube by 6. Find the side length with a cube root first, then work out the area of a face or the length of the edges from it.