GCSE Maths · Higher · Chapter 14: Powers and standard form · Lesson 1

Index laws

Grades 4–7About 45 minutesNo calculator38 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/gcse-index-laws/

NameClassDate

Do Now

Grades 4–6

The first question is what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Work out $(- 2)^{3}$ .
[1 mark]
Answer
A2
From chapter 10
Does $(4, 16)$ lie on $y = 5 x - 5$ ?
[1 mark]
Answer
A3
Last lesson
Two fair dice are rolled. How many outcomes are there?
[1 mark]
Answer
A4
Key Stage 3
Write $20 : 35$ in its simplest form.
[1 mark]
Answer

Grades 4–6

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

Example 1

Simplify $4 x^{5} \times 3 x^{2}$ .

Hint Multiply the numbers. Add the powers of

x

Example 2

Simplify $\frac{20 a ^{9}}{4 a ^{3}}$ .

Hint Divide the numbers. Subtract the powers of

a

Grades 5–6

C1
Work out the value of $4^{- 2}$ and the value of $9^{0}$ .
[2 marks]
Answer $4^{- 2} = \dots\dots$ $9^{0} = \dots\dots$
C2
Write $\frac{3 ^{7} \times 3 ^{2}}{3 ^{5}}$ as a single power of 3.
[2 marks]
Answer

C3
Simplify $(2 p^{4})^{3}$ .
[2 marks]
Answer
C4
$7^{2 n} \times 7^{3} = 7^{11}$ . Work out the value of $n$ .
[2 marks]
Answer
C5
Write $\frac{1}{125}$ as a power of 5.
[1 mark]
Answer
C6
Simplify $\frac{6 y ^{3}}{18 y ^{7}}$ . Give your answer as a fraction with no negative power.
[2 marks]
Answer
C7
Simplify $3 a^{2} b^{4} \times 5 a^{3} b$ .
[2 marks]
Answer
C8
Which is larger, $2^{15}$ or $4^{7}$ ? Show how you decide, without working out either one.
[2 marks]

Grades 6–7

D1
Jo says, " $3^{4} \times 3^{2} = 9^{6}$ , because you multiply the 3s and add the powers."
Explain why Jo is wrong, and write $3^{4} \times 3^{2}$ as a single power of 3.
[2 marks]
Answer

D2
A rectangle has length $8 x^{5}$ cm and width $2 x^{3}$ cm.
(a)
Write an expression for the area of the rectangle. Give your answer in its simplest form.
[2 marks]
Answer
(b)
A square has the same area as the rectangle. Write an expression for the side length of the square.
[2 marks]
Answer
(c)
The length of the rectangle is $k$ times its width. Write $k$ in terms of $x$ .
[2 marks]
Answer
Total for D2: 6 marks
D3
Write $\frac{( a ^{3} ) ^{4} \times a ^{- 2}}{a ^{5}}$ as a single power of $a$ .
[3 marks]
Answer

Grade 8 and beyond

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out zero and negative powers.
I can multiply, divide and raise powers, with numbers in front.
I can compare powers and solve for an unknown power.

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.