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In this worksheet, students multiply numbers with radical exponents.

Key stage:  KS 4

Curriculum topic:  Number

Curriculum subtopic:  Calculate with Roots and Integer (Fractional) Indice

Difficulty level:

### QUESTION 1 of 10

A radical is a square root, a cube root, a fourth root, a fifth root etc...

The symbol   n   is used to represent the nth root.

5√32 = 2 because 25 = 32

4√81 = 3 because 34 = 81

We can also write   4√81 as 81¼

because 81¼ x  81¼ x  81¼ x  81¼ =  811 = 81

This worksheet is about using the rule that

(ab)n = anbn

or its converse

anbn = (ab)n

and the rule that

ap/q = (a1/q)p

Example

Work out, without a calculator,

8¾ x  32¾

This is

(8¼ x  32¼)3

or (4√8 x 4√32)3

Now...

8¼ 32¼  =  (8 x 32)¼ =  256¼

4√256 = 4 because 44 = 256

Now we must cube the result to get 43 = 64

Work out, without a calculator,

8¾ x   2¾

Work out, without a calculator,

1¾ x   1¾

Work out, without a calculator,

33/2 x   33/2

Work out, without a calculator,

33/2 x   123/2

Work out, without a calculator,

23/2 x   323/2

Work out, without a calculator,

25/2 x   85/2

Work out, without a calculator,

27¾ x   3¾

Work out, without a calculator,

92/3 x   32/3

Work out, without a calculator,

125¾ x   5¾

Work out, without a calculator,

122/3 x   182/3

• Question 1

Work out, without a calculator,

8¾ x   2¾

8
EDDIE SAYS
(8 × 2)¼
= 16¼
= 4√16
= 2
2³ = 8
• Question 2

Work out, without a calculator,

1¾ x   1¾

1
EDDIE SAYS
(1 × 1)¼
= 1¼
= 4√1
= 1
1sup3; = 1
• Question 3

Work out, without a calculator,

33/2 x   33/2

27
EDDIE SAYS
(3 × 3)½
= 9½
= √9
= 3
3³ = 27
• Question 4

Work out, without a calculator,

33/2 x   123/2

216
EDDIE SAYS
(3 × 12)½
= 36½
= √36
= 6
6³ = 216
• Question 5

Work out, without a calculator,

23/2 x   323/2

512
EDDIE SAYS
(2 × 32)½
= 64½
= √64
= 8
8³ = 512
• Question 6

Work out, without a calculator,

25/2 x   85/2

1024
EDDIE SAYS
(2 × 8)½
= 16½
= √16
= 4
45 = 1024
• Question 7

Work out, without a calculator,

27¾ x   3¾

27
EDDIE SAYS
(27 × 3)¼
= 81¼
= 4√81
= 3
3³ = 27
• Question 8

Work out, without a calculator,

92/3 x   32/3

9
EDDIE SAYS
(9 × 3)1/3
= 271/3
= 3√27
= 3
3² = 9
• Question 9

Work out, without a calculator,

125¾ x   5¾

125
EDDIE SAYS
(125 × 5)¼
= 625¼
= 4√625
= 5
5³ = 125
• Question 10

Work out, without a calculator,

122/3 x   182/3

36
EDDIE SAYS
(12 × 18)1/3
= 2161/3
= 3√216
= 6
6² = 36
---- OR ----

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