A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 1: Algebraic expressions · Lesson 1

Index laws, including negative and fractional indices

74 marks

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Date

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Quick questions to start.

A1
Needed today
Evaluate $2 7^{\frac{2}{3}}$ .
[2 marks]
Answer
A2
GCSE Higher
Simplify $(3 a^{2} b)^{3}$ .
[2 marks]
Answer
A3
GCSE Higher
Expand and simplify $(2 x - 3) (x + 4)$ .
[2 marks]
Answer
A4
GCSE Higher
Solve $2^{x} = \frac{1}{8}$ .
[2 marks]
Answer $x = \dots\dots$

Example, then your turn

A-Level

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

Example 1

Simplify $\frac{( 4 x ^{5} y ^{2} ) ^{2}}{2 x ^{3} y}$ .

Hint Square the

4

, the

x^{5}

and the

y^{2}

separately. Then divide the numbers, and subtract the powers of

x

and of

y

B1
Simplify $(3 a^{4} b^{3})^{2} \times 2 a^{- 1} b$ .
[3 marks]
Answer

Example 2

Evaluate (a) $21 6^{- \frac{2}{3}}$ (b) $(\frac{8}{125})^{- \frac{2}{3}}$

Hint A negative power means one over. The bottom of the fraction in the power is the root; the top is the power. Take the root first: the numbers stay small.

B2
Evaluate (a) $12 8^{- \frac{3}{7}}$ (b) $(\frac{100}{9})^{- \frac{3}{2}}$
[4 marks]
Answer(a) $\dots\dots$ (b) $\dots\dots$

Example 3

Write $\frac{3 x ^{2} - 4 x}{2 x}$ in the form $a x^{p} + b x^{q}$ , where $a$ , $b$ , $p$ and $q$ are constants.

Hint Write

x

x^{\frac{1}{2}}

, then divide each term on top by

2 x

separately.

B3
Write $\frac{5 x ^{3} + 2 x}{x x}$ in the form $a x^{p} + b x^{q}$ .
[3 marks]
Answer

Example 4

Write $8^{x + 1} \times 4^{2 x}$ as a single power of $2$ .

Hint

8 = 2^{3}

and

4 = 2^{2}

. Then

(2^{3})^{x + 1} = 2^{3 x + 3}

B4
Write $\frac{9 ^{x}}{2 7 ^{x - 2}}$ as a single power of $3$ .
[3 marks]
Answer

Example 5

Solve $4^{x + 2} = 8^{2 x - 1}$ .

Hint Write both sides as powers of

2

. If

2^{m} = 2^{n}

then

m = n

B5
Solve $9^{x + 1} = \frac{2 7 ^{x}}{3}$ .
[3 marks]
Answer $x = \dots\dots$

Practice

A-Level

C1
Simplify $(2 x^{\frac{1}{2}})^{3} \times 3 x^{- \frac{1}{2}}$ .
[2 marks]
Answer

C2
Write $\frac{( 2 + x ) ^{2}}{x}$ in the form $a x^{- 1} + b x^{- \frac{1}{2}} + c$ , where $a$ , $b$ and $c$ are integers.
[3 marks]
Answer
C3
Given that $y = 3^{x}$ , write $9^{x + 1} - 3^{x - 1}$ in terms of $y$ .
[3 marks]
Answer
C4
Solve $(\frac{1}{16})^{x} = 8^{x + 2}$ .
[3 marks]
Answer $x = \dots\dots$

Exam-style questions

Edexcel exam style

D1
Given that $y = \frac{1}{8} x^{3}$ , express each of the following in the form $k x^{n}$ , where $k$ and $n$ are constants.
(a)
$y^{\frac{1}{3}}$
[2 marks]
Answer
(b)
$\frac{1}{2} y^{- \frac{2}{3}}$
[3 marks]
Answer
Total for D1: 5 marks

D2
$f (x) = \frac{( 2 x - 3 ) ^{2}}{x x}, x > 0$
(a)
Show that $f (x)$ can be written as $4 x^{- \frac{1}{2}} - 12 x^{- 1} + 9 x^{- \frac{3}{2}}$ .
[4 marks]
(b)
Hence find the exact value of $f (9)$ .
[2 marks]
Answer
(c)
Solve $f (x) = 0$ .
[2 marks]
Answer $x = \dots\dots$
Total for D2: 8 marks
D3
Ravi writes: $(\frac{4}{x ^{2}})^{- \frac{3}{2}} = 4 \times (- \frac{3}{2}) \times x^{3} = - 6 x^{3}$
Ravi is wrong.
(a)
Describe the two mistakes Ravi has made.
[2 marks]
(b)
Write $(\frac{4}{x ^{2}})^{- \frac{3}{2}}$ in the form $k x^{n}$ .
[3 marks]
Answer
(c)
Hence solve $(\frac{4}{x ^{2}})^{- \frac{3}{2}} = 27$ , where $x > 0$ .
[3 marks]
Answer $x = \dots\dots$
Total for D3: 8 marks
D4
Given that $y = 2^{x}$
(a)
write $4^{x}$ in terms of $y$ .
[1 mark]
Answer
(b)
Show that the equation $2^{2 x + 1} - 17 \times 2^{x} + 8 = 0$ can be written as $2 y^{2} - 17 y + 8 = 0$ .
[2 marks]
(c)
Hence solve $2^{2 x + 1} - 17 \times 2^{x} + 8 = 0$ .
[4 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
Total for D4: 7 marks
D5
Find the exact value of $x$ for which $\frac{9 ^{x}}{3 ^{1 - x}} = 273$ .
(a)
Write $273$ as a power of $3$ .
[2 marks]
Answer
(b)
Hence find $x$ .
[3 marks]
Answer $x = \dots\dots$
Total for D5: 5 marks

Extension

Stretch

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

E1
Find every real number $x$ for which $(x^{2} - 5 x + 5)^{x^{2} - 9 x + 20} = 1$ , and work out their sum.
[6 marks]
Hint 1 · What to try
A power equals $1$ in three ways: the base is $1$ ; the power is $0$ (and the base is not $0$ ); or the base is $- 1$ and the power is even.
Hint 2 · The first line
Base $1$ : $x^{2} - 5 x + 4 = 0$ . Power $0$ : $x^{2} - 9 x + 20 = 0$ . Base $- 1$ : $x^{2} - 5 x + 6 = 0$ , then check the power is even.
Hint 3 · The full method
Base $1$ gives $x = 1, 4$ . Power $0$ gives $x = 4, 5$ (the base is not $0$ there). Base $- 1$ gives $x = 2, 3$ , where the power is $6$ and $2$ , both even. So $x = 1, 2, 3, 4, 5$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the index laws with negative and fractional powers.
I can write an expression as a sum of powers of x.
I can write numbers as powers of one base to solve equations.
I can spot a quadratic in disguise when y = a^x.

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