KS3 Maths · Year 8 · Chapter 9: Decimal numbers · Lesson 3

Multiplying with numbers in standard form

Year 8About 60 minutesNo calculator48 marks

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Quick questions to start.

A1
Needed today
Write $1 0^{4} \times 1 0^{3}$ as a single power of $10$ .
[1 mark]
Answer
A2
Last lesson
Write $0.00058$ in standard form.
[1 mark]
Answer
A3
From chapter 8
Factorise $6 x + 15$ .
[1 mark]
Answer
A4
Year 7
Work out $3.5 \times 0.6$ .
[1 mark]
Answer

Example, then your turn

Year 8

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

Example 1

Work out each answer. Give your answers in standard form.

(a) $(2 \times 1 0^{4}) \times (3 \times 1 0^{5})$

(b) $(1.2 \times 1 0^{- 3}) \times (4 \times 1 0^{7})$

Hint Multiply the front numbers. Then multiply the powers of

10

by adding the indices.

B1
Work out each answer. Give your answers in standard form.
(a) $(4 \times 1 0^{3}) \times (2 \times 1 0^{6})$
(b) $(3.1 \times 1 0^{- 5}) \times (3 \times 1 0^{2})$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 2

Work out each answer. Give your answers in standard form.

(a) $(5 \times 1 0^{3}) \times (8 \times 1 0^{4})$

(b) $(6 \times 1 0^{- 2}) \times (7 \times 1 0^{5})$

Hint The front numbers make

40

and

42

, which are not between

1

and

10

. Write

40

4 \times 1 0^{1}

B2
Work out each answer. Give your answers in standard form.
(a) $(9 \times 1 0^{6}) \times (4 \times 1 0^{2})$
(b) $(2.5 \times 1 0^{4}) \times (8 \times 1 0^{- 7})$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 3

Work out each answer. Give your answers in standard form.

(a) $(3 \times 1 0^{- 4}) \times (2 \times 1 0^{- 3})$

(b) $(4.5 \times 1 0^{- 2}) \times (6 \times 1 0^{- 5})$

Hint Adding two negative indices gives a more negative index:

- 4 + - 3 = - 7

B3
Work out each answer. Give your answers in standard form.
(a) $(2 \times 1 0^{- 6}) \times (4 \times 1 0^{- 2})$
(b) $(7.5 \times 1 0^{- 3}) \times (4 \times 1 0^{- 4})$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 4

Work out each answer. Give your answers in standard form.

(a) $(3 \times 1 0^{4})^{2}$

(b) $(2 \times 1 0^{- 3})^{3}$

Hint

(3 \times 1 0^{4})^{2}

means

(3 \times 1 0^{4}) \times (3 \times 1 0^{4})

. Square the front number and double the index.

B4
Work out each answer. Give your answers in standard form.
(a) $(6 \times 1 0^{5})^{2}$
(b) $(5 \times 1 0^{- 2})^{3}$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 5

A grain of salt has a mass of $6 \times 1 0^{- 5}$ grams. A packet holds $2.5 \times 1 0^{7}$ grains. Work out the mass of salt in the packet, in grams. Give your answer in standard form.

Hint Multiply the mass of one grain by the number of grains.

B5
A bacterium is $2.5 \times 1 0^{- 6}$ m long. $4 \times 1 0^{7}$ of these bacteria are laid end to end in a line. Work out the length of the line, in metres. Give your answer in standard form.
[3 marks]
Answer

Practice

Year 8

C1
Write down the missing number. Give it in standard form.
$(3 \times 1 0^{4}) \times □ = 1.2 \times 1 0^{9}$
[2 marks]
Answer
C2
Work out each answer. Give your answers in standard form, correct to $2$ significant figures.
(a) $(2.4 \times 1 0^{3}) \times (3.7 \times 1 0^{5})$
(b) $(1.9 \times 1 0^{- 2}) \times (4.3 \times 1 0^{- 3})$
[2 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

C3
Work out $2.5 \times 1 0^{- 3} \times 60000$ . Give your answer in standard form.
[2 marks]
Answer
C4
A field is a rectangle $4 \times 1 0^{3}$ m long and $2.5 \times 1 0^{2}$ m wide. Work out its area, in m². Give your answer in standard form.
[2 marks]
Answer
C5
Which is larger, P or Q? By how much is it larger?
P $= (6 \times 1 0^{4}) \times (5 \times 1 0^{- 3})$ Q $= (7 \times 1 0^{- 2}) \times (4 \times 1 0^{3})$
[2 marks]
Answer

Exam-style questions

GCSE Foundation

D1
Work out each answer. Give your answers in standard form.
(a)
$(2 \times 1 0^{5}) \times (4 \times 1 0^{3})$
[1 mark]
Answer
(b)
$(5 \times 1 0^{- 3}) \times (3 \times 1 0^{- 2})$
[1 mark]
Answer
(c)
$(7 \times 1 0^{4})^{2}$
[1 mark]
Answer
Total for D1: 3 marks

D2
Leo works out $(4 \times 1 0^{5}) \times (3 \times 1 0^{2})$ . He writes $12 \times 1 0^{10}$ .
Leo has made two mistakes. Explain each one. Write the correct answer in standard form.
[3 marks]
D3
Radio signals travel at $3 \times 1 0^{8}$ metres per second. A signal from a space probe takes $4 \times 1 0^{3}$ seconds to reach the Earth.
(a)
Work out the distance from the probe to the Earth, in metres. Give your answer in standard form.
[2 marks]
Answer
(b)
Write this distance in kilometres. Give your answer in standard form.
[2 marks]
Answer
Total for D3: 4 marks
D4
A factory packs bolts and washers in boxes. The table shows how many are in each box and the mass of one.
(a)
Work out the mass of the bolts in a box, in kg. Give your answer in standard form.
[2 marks]
Answer
(b)
Work out the mass of the washers in a box, in kg. Give your answer in standard form.
[2 marks]
Answer
(c)
Which box is heavier, and by how many kilograms?
[1 mark]
Answer
Total for D4: 5 marks
D5
A square floor tile has sides of length $6 \times 1 0^{- 2}$ m. A hall floor is covered with $5 \times 1 0^{4}$ of these tiles. Work out the area of the floor, in m². Give your answer in standard form.
[4 marks]
Answer

Extension

Stretch

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

E1
$(4 \times 1 0^{n}) \times (5 \times 1 0^{3}) = 2 \times 1 0^{- 5}$
Work out the value of $n$ .
[4 marks]
Hint 1 · What to try
Multiply the front numbers first. What do $4$ and $5$ make?
Hint 2 · The first line
$4 \times 5 = 20 = 2 \times 1 0^{1}$ , so the left side is $2 \times 1 0^{1 + n + 3}$ .
Hint 3 · The full method
The powers must match: $1 + n + 3 = - 5$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can multiply numbers in standard form by multiplying the front numbers and adding the powers.
I can put an answer such as $42 \times 1 0^{3}$ back into standard form.
I can square a number in standard form and use multiplying in a problem.

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