AQA Level 2 Further Maths · Level 2 Certificate · Chapter 4: Algebra IV · Lesson 5

Indices

About 80 minutesNo calculator76 marks

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Quick questions to start.

A1
GCSE Higher
Work out the exact value of $(\frac{16}{9})^{- \frac{3}{2}}$ .
[2 marks]
Answer
A2
Needed today
Solve $x^{2} - 10 x + 21 = 0$ .
[2 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
A3
Last lesson
Solve $2 x^{2} - 7 x - 4 ⩽ 0$ .
[2 marks]
Answer $\dots\dots\dots\dots$
A4
From chapter 3
The graph of $y = a \times 3^{x}$ passes through the point $(2, 36)$ . Work out the value of $a$ .
[2 marks]
Answer $a = \dots\dots$

Example, then your turn

Level 2

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

Example 1

Write $\frac{3 x ^{2} \times x ^{4}}{( x ^{2} ) ^{- \frac{1}{3}}}$ as a single power of $x$ .

Hint Write every part as a power of

x

first: a cube root is the power

\frac{1}{3}

. Then add the indices on top and subtract the index underneath.

B1
Write $\frac{( x ^{3} ) ^{\frac{1}{4}} \times 4 x ^{5}}{x ^{- \frac{1}{2}}}$ as a single power of $x$ .
[3 marks]
Answer $\dots\dots\dots\dots$

Example 2

Solve $x^{- \frac{5}{2}} = \frac{1}{243}$

Hint Take the reciprocal of both sides to make the index positive. Then undo the power

\frac{5}{2}

one step at a time.

B2
Solve $x^{- \frac{3}{4}} = \frac{8}{27}$
[3 marks]
Answer $x = \dots\dots$

Example 3

Expand and simplify $(x^{\frac{1}{2}} + 3 x^{- \frac{1}{2}})^{2}$

Hint Write it as two brackets multiplied. When you multiply powers of

x

, add the indices:

x^{\frac{1}{2}} \times x^{- \frac{1}{2}} = x^{0} = 1

B3
Expand and simplify $(2 x^{\frac{1}{2}} - x^{- \frac{1}{2}}) (x^{\frac{1}{2}} + 3 x^{- \frac{1}{2}})$
[3 marks]
Answer $\dots\dots\dots\dots$

Example 4

Solve $x^{4} - 11 x^{2} + 18 = 0$ . Give your answers in exact form.

Hint There are no odd powers of

x

. Let

y = x^{2}

, so

x^{4} = y^{2}

B4
Solve $2 x^{4} + 5 x^{2} - 12 = 0$ . Give your answers in exact form.
[4 marks]
Answer $x = \dots\dots\dots\dots$

Example 5

Solve $9^{x} - 6 \times 3^{x} - 27 = 0$

Hint

9^{x} = (3^{2})^{x} = (3^{x})^{2}

, so let

y = 3^{x}

. Remember that

3^{x}

can never be negative.

B5
Solve $4^{x + 1} - 33 \times 2^{x} + 8 = 0$
[5 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$

Practice

Level 2

C1
Write both sides as powers of the same number. Then solve each equation.
(a)
$8^{x} = 4^{x + 3}$
[2 marks]
Answer $x = \dots\dots$
(b)
$2 5^{x} = \frac{1}{125}$
[2 marks]
Answer $x = \dots\dots$
Total for C1: 4 marks
C2
Solve $2 x - 9 x + 4 = 0$
[4 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$

C3
Solve $x^{\frac{2}{3}} - 5 x^{\frac{1}{3}} + 4 = 0$
[4 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
C4
Solve the simultaneous equations
$2^{x} \times 4^{y} = 32 and 3^{x} \div 9^{y} = \frac{1}{27}$
[4 marks]
Answer $x = \dots\dots$ , $y = \dots\dots$
C5
Solve $x^{6} + 7 x^{3} - 8 = 0$
[4 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
C6
Write $\frac{3 x ^{2} - 5 x ^{\frac{1}{2}}}{x ^{\frac{3}{2}}}$ in the form $a x^{p} + b x^{q}$
[3 marks]
Answer $\dots\dots\dots\dots$

Exam-style questions

AQA exam style

D1
$\frac{y ^{\frac{5}{2}} \times y ^{\frac{1}{2}}}{y ^{- 1}} = y^{m}$
Circle the value of $m$ .
$234 \frac{5}{4}$
[1 mark]

D2
$a^{n} \times (a^{3})^{n} = \frac{a ^{20}}{a ^{n}}$
Work out the value of $n$ .
[2 marks]
Answer $n = \dots\dots$
D3
Ali says, " $9^{x}$ is the same as $3 \times 3^{x}$ ."
Ali is wrong.
(a)
Show that Ali is wrong.
[1 mark]
(b)
Solve $9^{x} + 3^{x} = 90$ . You must show your working.
[4 marks]
Answer $x = \dots\dots$
Total for D3: 5 marks
D4
$\frac{x ^{2}}{2} + \frac{8}{x ^{2}} = 5$
(a)
Show that the equation can be written as $x^{4} - 10 x^{2} + 16 = 0$
[2 marks]
(b)
Hence solve $\frac{x ^{2}}{2} + \frac{8}{x ^{2}} = 5$ . Give your answers in exact form.
[3 marks]
Answer $x = \dots\dots\dots\dots$
Total for D4: 5 marks
D5
Solve $2^{x^{2}} \times 4^{x - 3} = 8^{x}$
You must show your working.
[5 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$
D6
Solve $2 x^{\frac{1}{2}} + 15 x^{- \frac{1}{2}} = 11$
[4 marks]
Answer $x = \dots\dots$ or $x = \dots\dots$

Extension

Stretch

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

E1
Find every value of $x$ for which $(x^{2} - 7 x + 11)^{x - 2} = 1$
[5 marks]
Hint 1 · What to try
A power $a^{b}$ equals $1$ in three ways: $a = 1$ ; or $b = 0$ (with $a \neq = 0$ ); or $a = - 1$ and $b$ is even.
Hint 2 · The first line
Base $1$ : $x^{2} - 7 x + 10 = 0$ . Index $0$ : $x = 2$ . Base $- 1$ : $x^{2} - 7 x + 12 = 0$ , then check the index is even.
Hint 3 · The full method
Base $1$ gives $x = 2$ or $5$ . Index $0$ gives $x = 2$ again. Base $- 1$ gives $x = 3$ (index $1$ , odd: rejected) or $x = 4$ (index $2$ , even). So $x = 2$ , $4$ or $5$ .
Answer $x = \dots\dots\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can use the index laws with fractional and negative powers.
I can solve an equation with a fractional index.
I can solve a quadratic in disguise.
I can solve an equation with the unknown in the index.

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