KS3 Maths · Year 7 · Chapter 1: Working with numbers · Lesson 4

Powers and roots

Year 7About 60 minutesNo calculator49 marks

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Date

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Quick questions to start.

A1
Needed today
Work out $14 \times 14$ .
[1 mark]
Answer
A2
Needed today
Work out $0.3 \times 0.3$ .
[1 mark]
Answer
A3
Last lesson
Find the LCM of $9$ and $12$ .
[1 mark]
Answer
A4
From chapter 1
Work out $- 40 \div 8$ .
[1 mark]
Answer

Example, then your turn

Year 7

Your teacher works through each example with you. Then try the one beside it.

Example 1

(a) Write $2 \times 2 \times 2 \times 2 \times 2 \times 2$ as a power of $2$ .

(b) Work out $5^{4}$ .

Hint The power tells you how many of the number are multiplied together.

B1
(a) Write $9 \times 9 \times 9 \times 9$ as a power of $9$ .
(b) Work out $2^{8}$ .
(c) Work out $0. 5^{2}$ .
[3 marks]
Answer(a) $\dots$ (b) $\dots$ (c) $\dots$

Example 2

Work out

(a) $(- 3)^{2}$ (b) $- 3^{2}$ (c) $(- 2)^{5}$

Hint A bracket means the whole of

- 3

is squared. With no bracket, only the

3

is squared.

B2
Work out
(a) $(- 5)^{3}$
(b) $- 4^{2}$
(c) $(- 10)^{4}$
[3 marks]
Answer(a) $\dots$ (b) $\dots$ (c) $\dots$

Example 3

(a) Find both values of $x$ when $x^{2} = 144$ .

(b) Work out $0.49$ .

Hint A square root undoes squaring. A negative number squared is positive too.

B3
(a) Find both values of $x$ when $x^{2} = 324$ .
(b) Work out $2.25$ .
[2 marks]
Answer(a) $\dots$ (b) $\dots$

Example 4

Work out

(a) $38000$ (b) $3 - 1000$

Hint Find the number that, multiplied by itself three times, gives the number under the root.

B4
Work out
(a) $327000$ (b) $3 - 125$
[2 marks]
Answer(a) $\dots$ (b) $\dots$

Example 5

A square has an area of $900 cm^{2}$ . Work out its perimeter.

Hint The side length is the square root of the area.

B5
A cube has a volume of $512 cm^{3}$ . Work out the area of one face of the cube.
[3 marks]
Answer

Practice

Year 7

C1
Work out $4^{3} - 8^{2}$ .
[2 marks]
Answer
C2
Which is larger, $2^{9}$ or $9^{2}$ ? Work out the difference between them.
[2 marks]
Answerlarger $\dots$ difference $\dots$

C3
Find the value of $n$ when $5^{n} = 625$ .
[2 marks]
Answer
C4
$70$ lies between two whole numbers next to each other. Write down the two whole numbers.
[2 marks]
Answer
C5
$61$ is the sum of two square numbers. Find the two square numbers.
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Work out the value of each of these.
(a)
$6^{2} + 2^{3}$
[1 mark]
Answer
(b)
$5^{2} + 1 2^{2}$
[2 marks]
Answer
(c)
$(- 2)^{3} \times 3^{2}$
[2 marks]
Answer
Total for D1: 5 marks

D2
Amy says, " $4^{3} = 12$ , because $4 \times 3 = 12$ ."
Amy is wrong. Explain why, and write down the correct value of $4^{3}$ .
[2 marks]
D3
A storage box is a cube. Its volume is $343 cm^{3}$ . Work out the total surface area of the box.
[4 marks]
Answer
D4
A square garden has an area of $196 m^{2}$ . A fence is put all the way round the edge of the garden. The fence costs £ $12$ per metre. Work out the cost of the fence.
[4 marks]
Answer
D5
The number $64$ can be written as a power of several different whole numbers.
(a)
Write $64$ as a power of $2$ .
[1 mark]
Answer
(b)
Write $64$ as a power of $4$ .
[1 mark]
Answer
(c)
Write $64$ as a power of $8$ .
[1 mark]
Answer
Total for D5: 3 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
$1^{3} + 2^{3} + 3^{3} + 4^{3} = 100$ , and $100$ is a square number.
Work out $1^{3} + 2^{3} + 3^{3} + \dots + 1 0^{3}$ without adding up the ten cubes.
[4 marks]
Hint 1 · What to try
Work out $1^{3}$ , then $1^{3} + 2^{3}$ , then $1^{3} + 2^{3} + 3^{3}$ . Each total is a square number. Write down the number that is squared each time.
Hint 2 · The first line
The numbers being squared are $1$ , $3$ , $6$ , $10$ . These are $1$ , $1 + 2$ , $1 + 2 + 3$ , $1 + 2 + 3 + 4$ .
Hint 3 · The full method
So the sum of the first ten cubes is $(1 + 2 + \dots + 10)^{2} = 5 5^{2} = 3025$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out powers, including powers of negative numbers.
I can find square roots and cube roots without a calculator.
I can find a side length from the area of a square or the volume of a cube.

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