1. Fractional Indices

2. Starter Simplify the following… a) 4-2 b) 2-4 c) 3-8
d) e) f) 4-2 x 4-3
Find the value of…
a) b) c) 3-2
d) e) f) 6-2 x 63
1/42
1/24
1/38
1/5 1
1/45
1/64
1/32
1/9
1/125
1/4 6

3. Fractional Indices We have seen how to deal with Negative Indices
We saw that a Negative power turns the term into a Fraction
Today we will see how to work out what a Fractional Power means

4. Fractional Indices 4 x 4 = 4 2 x 2 = 4 4 = 2
Remember when multiplying, we add the powers…
1 2
1 2 1 4 x 4 = 4
These numbers must be the same 2 x 2 = 4
1 2 4 = 2
Therefore

5. Fractional Indices 9 x 9 = 9 3 x 3 = 9 9 = 3
Remember when multiplying, we add the powers…
1 2
1 2 1 9 x 9 = 9
These numbers must be the same 3 x 3 = 9
1 2 9 = 3
Therefore

6. These numbers must be the same
Fractional Indices
1 2
1 2 1 25 x 25 = 25
These numbers must be the same 5 x 5 = 25
1 2 25 = 5
Therefore

7. You can leave them as surds
Fractional Indices
1 2 4 = 2
‘If something is raised to the power 1/2 , you work out the square root’
1 2 9 = 3
1 2 25 = 5
1 2
You can leave them as surds 2 = √2

8. These numbers must be the same
Fractional Indices
1 3
1 3
1 3 1 8 x 8 x 8 = 8
These numbers must be the same 2 x 2 x 2 = 8
1 3 8 = 2
Therefore

9. These numbers must be the same
Fractional Indices
1 3
1 3
1 3 1
1000 x
1000 x
1000 =
1000
These numbers must be the same 10 x 10 x 10 =
1000
1 3
1000 = 10
Therefore

10. Fractional Indices 8 = 2 1000 = 10 27 = 3 5 =
1 3 8 = 2
‘If something is raised to the power 1/3, you work out the cube root’
1 3
1000 = 10
1 3 27 = 3
1 3
You can leave them as surds 5 =
3√5

11. Fractional Indices
1 2 x = √x
1 3 x =
3√x
1 4 x =
4√x
1 m x =
m√x

12. Fractional Indices 8 x 8 = 8 2 x 2 = 4 8 = 4 1 3 1 3 2 3
Work out the cube roots 2 x 2 = 4
2 3 8 = 4
Therefore

13. Fractional Indices 27 x 27 = 27 3 x 3 = 9 27 = 9 1 3 1 3 2 3
Work out the cube roots 3 x 3 = 9
2 3 27 = 9
Therefore

14. Fractional Indices 10 x 10 = 10 x = (3√10)2 (3√10)2 10 = 3√10 3√10 1 3
2 3 10 x 10 = 10
Write as a surd
3√10 x
3√10 =
(3√10)2
2 3
(3√10)2 10 =
Therefore

15. Fractional Indices 7 x 7 x 7 = 7 x x = (5√7)3 (5√7)3 7 = 5√7 5√7 5√7
1 5
1 5
1 5
3 5 7 x 7 x 7 = 7
Write as surds x x
5√7
5√7
5√7 =
(5√7)3
3 5
(5√7)3 7 =
Therefore

16. Fractional Indices 10 = (3√10)2 7 = (5√7)3 2 = (7√2)4 x = (m√x)n 2 3
3 5 7 =
(5√7)3
4 7 2 =
(7√2)4
n m x =
(m√x)n

17. Plenary
= (3√8)4
= (2)4 16

18. Plenary
1/100 x 8
0.01 x 8
0.08 23
(3√x)2
= x
(3√8)2
= 4

19. Summary We have learnt how to deal with Fractional Indices
We have also seen how to rewrite them when both the numerator and denominator are not equal to 1