A2400 ch1f | Version 1.1 | September 2020

Simplifying indices

A LEVEL LINKS Scheme of work: 1a. Algebraic expressions – basic algebraic manipulation, indices and surds

Key points • am × an = am + n

•

• (am)n = amn • a0 = 1

• i.e. the nth root of a

•

•

• The square root of a number produces two solutions, e.g. .

Example 1 Simplify

= 3x3 6 ÷ 2 = 3 and use the rule to

give

Example 2 Simplify

= x8 − 4 = x4

1 Use the rule

2 Use the rule

Example 3 Write as a single power of x

mm n

n

a aa

-=

1nna a=

( )m mn m nna a a= =

1mmaa

- =

16 4= ±

5

262xx

5

262xx

mm n

na aa

-=

55 2 3

2x x xx

-= =

3 5

4x xx´

3 5 3 5 8

4 4 4x x x xx x x

+´= = m n m na a a +´ =

mm n

na aa

-=

13x

A2400 ch1f | Version 1.1 | September 2020

Use the rule , note that the

fraction remains unchanged

Example 4 Write as a single power of x

1 Use the rule

2 Use the rule

Practice questions

1 Simplify.

a b

c d

e f

g h

2 Write the following as a single power of x.

a b c

d e f

3 Write the following without negative or fractional powers.

a b x0 c

d e f

11 13 3

xx

-= 1 mm aa

-=

13

4x

12

12

4 4

4

x x

x-

=

=

1nna a=

1 mm aa

-=

2 3

232x xx´ 5

2102x

x x´3

33 22x xx´ 3 2

5714x yx y

12

2yy y´

12

322

cc c´

( )32

0

2

4

x

x

312 2

2 3x xx x-

´´

1x 7

1x

4 x

5 2x3

1x 3 2

1

x

3x-15x

25x

12x

-34x

-

Watch out!

Remember that any value raised to the power of zero is 1. This is the rule a

0 = 1.

A2400 ch1f | Version 1.1 | September 2020

Answers

1 a b 5x2

c 3x d

e f c–3

g 2x6 h x

2 a x–1 b x–7 c

d e f

3 a b 1 c

d e f

332x

22yx

12y

14x

25x

13x

-23x

-

31x

5 x

5 2x 1x 4 3

1

x