GCSE Maths · Higher · Chapter 16: Counting, accuracy, powers and surds · Lesson 2

Negative and fractional powers

Grades 5–8About 45 minutesNo calculator37 marks

Hints online, answers free with an account:
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NameClassDate

Do Now

Grades 4–6

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

A1
Needed today
Write down the value of $9 6^{0}$ .
[1 mark]
Answer
A2
Last lesson
Write $0. 5 ˙$ as a fraction.
[1 mark]
Answer
A3
From chapter 6
Find the interior angle of a regular 9-sided polygon.
[1 mark]
Answer
A4
Key Stage 3
Find the area of a triangle with base $8$ cm and height $10$ cm.
[1 mark]
Answer

Grades 6–7

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

Example 1

Work out $4 9^{- \frac{3}{2}}$ .

Hint The bottom of the fraction is the root, the top is the power, and the minus sign turns it over.

Example 2

Work out $(\frac{4}{9})^{- \frac{1}{2}}$ .

Hint Turn the fraction over first, then take the square root of the top and of the bottom.

Grades 6–7

C1
Between which two consecutive whole numbers does $300$ lie?
[2 marks]
Answer $\dots\dots$ and $\dots\dots$
C2
Work out $6^{- 2}$ . Give your answer as a fraction.
[1 mark]
Answer

C3
Work out $62 5^{\frac{3}{4}}$ .
[2 marks]
Answer
C4
Find the value of $n$ when $2^{n} = \frac{1}{64}$ .
[2 marks]
Answer $n = \dots\dots$
C5
Write $\frac{1}{3 x ^{2}}$ in the form $x^{n}$ .
[2 marks]
Answer $\dots\dots$
C6
Simplify $(64 x^{9})^{\frac{2}{3}}$ .
[2 marks]
Answer
C7
Write $\frac{1}{32}$ as a power of 8.
[2 marks]
Answer $8^{\dots\dots}$
C8
Work out $5^{0} + 4^{- \frac{1}{2}}$ .
[2 marks]
Answer

Grades 7–8

D1
Leo says, " $8^{\frac{4}{3}} = \frac{32}{3}$ , because $8 \times \frac{4}{3} = \frac{32}{3}$ ."
Explain why Leo is wrong, and work out the correct value of $8^{\frac{4}{3}}$ .
[2 marks]

D2
$p = 24 3^{\frac{2}{5}}$ and $q = 24 3^{- \frac{3}{5}}$ .
(a)
Work out the value of $p$ .
[2 marks]
Answer
(b)
Work out the value of $q$ . Give your answer as a fraction.
[2 marks]
Answer
(c)
Write $p \div q$ as a power of 3.
[2 marks]
Answer
Total for D2: 6 marks
D3
A cube has a volume of $400$ cm $^{3}$ . Without a calculator, show that the length of an edge of the cube is between 7 cm and 8 cm.
[2 marks]

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 find the two whole numbers a root lies between.
I can work out a negative power as a reciprocal.
I can work out a fractional power as a root and a power, with numbers or letters.

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