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

Index laws, zero and negative powers, and standard form

Grades 4–9About 80 minutesNo calculator91 marks

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Before you start

Grades 4–5

You will need these in every question below.

A1
Work out $2^{5}$ .
[1 mark]
Answer

A2
Work out $(- 3)^{3}$ .
[1 mark]
Answer
A3
Work out $0.06 \times 10000$ .
[1 mark]
Answer
A4
Work out $5^{2} + 2^{3}$ .
[1 mark]
Answer
A5
Work out $7.2 \div 0.08$ .
[2 marks]
Answer

Examples, then your turn

Grades 5–6

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

Example 1

Write $3^{4} \times 3^{5}$ as a single power of 3.

Hint

3^{4}

is four 3s multiplied together and

3^{5}

is five more. Count them all: add the indices.

B1
Write $7^{6} \times 7^{3} \div 7^{4}$ as a single power of 7.
[2 marks]
Answer

Example 2

Work out the value of $6^{0}$ and the value of $2^{- 3}$ .

Hint Any non-zero number to the power 0 is 1. A negative index means one over the positive power:

2^{- 3} = \frac{1}{2 ^{3}}

B2
Work out the value of $5^{- 2} \times 5^{4} \div 5^{0}$ .
[2 marks]
Answer

Example 3

Simplify $(3 x^{4} y)^{2}$ .

Hint Square the number, and double every index:

(3 x^{4} y)^{2} = 3^{2} \times x^{4 \times 2} \times y^{2}

B3
Simplify $\frac{12 a ^{7} b ^{3}}{4 a ^{2} b}$ .
[3 marks]
Answer

Example 4

Write $0.000 52$ in standard form. Then write $3.07 \times 1 0^{5}$ as an ordinary number.

Hint Standard form is

A \times 1 0^{n}

with

1 \leq A < 10

. Count how many places the digits move.

B4
Work out $(4 \times 1 0^{5}) \times (6 \times 1 0^{- 2})$ . Give your answer in standard form.
[3 marks]
Answer

Practice

Grades 5–7

Each question asks for something different. Show your working.

C1
Use standard form in each part.
(a)
Write $72400000$ in standard form.
[1 mark]
Answer
(b)
Write $6.05 \times 1 0^{- 3}$ as an ordinary number.
[1 mark]
Answer
(c)
Write $38 \times 1 0^{4}$ in standard form.
[1 mark]
Answer
Total for C1: 3 marks

C2
Write $\frac{5 ^{8} \times 5 ^{- 3}}{5 ^{2}}$ as a single power of 5.
[2 marks]
Answer
C3
$3^{n} \times 3^{4} = 3^{- 2}$ . Work out the value of $n$ .
[2 marks]
Answer
C4
Simplify $(2 a^{3} b^{- 2})^{4}$ .
[2 marks]
Answer
C5
Write these numbers in order of size. Start with the smallest.
$4.3 \times 1 0^{- 2}$ $0.0405$ $4.12 \times 1 0^{- 2}$ $0.004 39$
[2 marks]
Answer
C6
Work these out. Give each answer in standard form.
(a)
$(8.4 \times 1 0^{6}) \div (2.1 \times 1 0^{- 3})$
[2 marks]
Answer
(b)
$(3.2 \times 1 0^{4}) + (5.9 \times 1 0^{3})$
[2 marks]
Answer
Total for C6: 4 marks
C7
A field is a rectangle. Work out the area of the field. Give your answer in standard form.
[2 marks]
Not drawn accurately
Answer
C8
Write $4^{3} \times 8^{2}$ as a single power of 2.
[2 marks]
Answer
C9
A sheet of paper is $8 \times 1 0^{- 3}$ cm thick.
(a)
Work out the height of a pile of 500 sheets, in cm.
[2 marks]
Answer
(b)
A pile of these sheets is 1.2 m high. Work out the number of sheets in the pile. Give your answer in standard form.
[2 marks]
Answer
Total for C9: 4 marks

Exam-style questions

Grades 6–9

Answer every question in the space given. Show your working: most of the marks are for method.

D1
Mo says, " $(4 \times 1 0^{3}) \times (5 \times 1 0^{4}) = 20 \times 1 0^{7}$ , and this answer is in standard form."
Explain why Mo is wrong, and write the answer in standard form.
[2 marks]
Answer

D2
A grain of sand has a mass of $6 \times 1 0^{- 4}$ grams.
(a)
Work out the mass of $2 \times 1 0^{5}$ grains of sand, in grams. Give your answer in standard form.
[2 marks]
Answer
(b)
Work out the number of grains of sand in 3 kg of sand. Give your answer in standard form.
[3 marks]
Answer
Total for D2: 5 marks
D3
Solve $\frac{9 ^{x}}{3 ^{x - 1}} = 2 7^{2}$ .
[4 marks]
Answer
D4
Show that $\frac{9 ^{4} \times 27}{3 ^{7}} = 3^{4}$ .
[3 marks]
D5
A country has a population of $4.5 \times 1 0^{7}$ people. On average, each person uses $1.2 \times 1 0^{2}$ litres of water a day.
(a)
Work out the total amount of water used in the country in one day, in litres. Give your answer in standard form.
[2 marks]
Answer
(b)
$1$ m³ is $1000$ litres. Write your answer to part (a) in m³.
[1 mark]
Answer
(c)
A reservoir holds $2.7 \times 1 0^{8}$ m³ of water. How many days of water use is this?
[2 marks]
Answer
Total for D5: 5 marks
D6
$8^{x} \times 4^{y} = 2^{13}$ and $\frac{9 ^{x}}{3 ^{y}} = 81$ .
Work out the value of $x$ and the value of $y$ . You must show your working.
[6 marks]
Answer $x = \dots\dots$ $y = \dots\dots$
D7
Planet $A$ is $1.2 \times 1 0^{8}$ km from the Sun. Planet $B$ is $7.8 \times 1 0^{8}$ km from the Sun.
(a)
How many times further from the Sun is planet $B$ than planet $A$ ?
[2 marks]
Answer
(b)
Light travels $3 \times 1 0^{5}$ km in one second. Work out the time light takes to travel from the Sun to planet $A$ , in seconds.
[2 marks]
Answer
Total for D7: 4 marks
D8
Simplify fully $(\frac{a ^{- 2} b ^{3}}{a ^{4} b ^{- 1}})^{2}$ .
Give your answer without negative indices.
[4 marks]
Answer
D9
Work out $\frac{3 \times 1 0 ^{5}}{5 \times 1 0 ^{- 2}} + 4 \times 1 0^{6}$ . Give your answer in standard form.
[3 marks]
Answer
D10
$n$ is a positive whole number.
(a)
Show that $2^{n} \times 5^{n + 2} = 25 \times 1 0^{n}$ .
[2 marks]
(b)
Hence write $2^{12} \times 5^{14}$ in standard form.
[1 mark]
Answer
Total for D10: 3 marks

Extension

Grade 9 and beyond

No route is given. Spend five minutes on a problem before you look at a hint. The hints are on the last page of the worksheet, and you can use one at a time.

E1
Solve $2^{x} \times 4^{x + 1} = 12 8^{2}$ .
[4 marks]
Hint 1 · What to try
Write every number as a power of 2.
Hint 2 · The first line
$4^{x + 1} = 2^{2 x + 2}$ and $12 8^{2} = (2^{7})^{2} = 2^{14}$ .
Hint 3 · The full method
So $x + 2 x + 2 = 14$ , which gives $3 x = 12$ and $x = 4$ .
Answer

E2
Show that $3^{200}$ is greater than $2^{300}$ .
[3 marks]
Hint 1 · What to try
Both indices are multiples of 100. Write each number as something to the power 100.
Hint 2 · The first line
$3^{200} = (3^{2})^{100}$ and $2^{300} = (2^{3})^{100}$ .
Hint 3 · The full method
So $3^{200} = 9^{100}$ and $2^{300} = 8^{100}$ . Since $9 > 8$ , $9^{100} > 8^{100}$ .
E3
The number $5^{20} \times 2^{23}$ is written out in full as an ordinary number. How many digits does it have?
[3 marks]
Hint 1 · What to try
Pair up each 5 with a 2: $5 \times 2 = 10$ .
Hint 2 · The first line
$5^{20} \times 2^{23} = 5^{20} \times 2^{20} \times 2^{3} = 1 0^{20} \times 8$ .
Hint 3 · The full method
$8 \times 1 0^{20}$ is 8 followed by twenty zeros, so it has 21 digits.
Answer
E4
$k$ is a positive whole number. Write $(4 \times 1 0^{k})^{3}$ in standard form. Give your answer in terms of $k$ .
[3 marks]
Hint 1 · What to try
Cube each part of the bracket separately.
Hint 2 · The first line
$4^{3} = 64$ and $(1 0^{k})^{3} = 1 0^{3 k}$ , so the value is $64 \times 1 0^{3 k}$ .
Hint 3 · The full method
$64$ is not between 1 and 10: $64 = 6.4 \times 10$ , so the answer is $6.4 \times 1 0^{3 k + 1}$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can multiply, divide and raise powers of the same number, using the index laws.
I can work out zero and negative powers, and simplify expressions with letters and indices.
I can write numbers in standard form and calculate with them without a calculator.

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