1. Laws of Rational Indices

2. Do you remember what a square root means?
If x2 = a, then
x is a square root of a.
Example:
42 = 16
4 is a square root of 16.
(4)2 = 16
4 is a square root of 16.
Both 4 and 4 are the square roots of 16.

3. How about if the index is not equal to 2?
If x3 = a
x is the cube root of a.
If x4 = a
x is a fourth root of a.
If x5 = a
x is the fifth root of a.
● ● ●
If xn = a
x is an nth root of a.

4. if xn = a, then x is an n th root of a.
In general, for any positive integer n,
if xn = a, then x is an n th root of a.
Radical n a
denote
n th root of a
‘ ’ is the radical sign. n

5. a In general, for any positive integer n,
if xn = a, then x is an n th root of a.
Radical n a
denote
n th root of a
Example:
43 = 64
64 = 4 3
The cube root of 64 is 4.
24 = 16
16 = 2 4
A fourth root of 16 is 2.
16 = 2 4 
(2)4 = 16
Another fourth root of 16 is 2.

6. a In general, for any positive integer n,
if xn = a, then x is an n th root of a.
Radical n a
denote
n th root of a
The ‘’ sign denotes the negative 4th root.
Example:
43 = 64
64 = 4 3
The cube root of 64 is 4.
24 = 16
16 = 2 4
A fourth root of 16 is 2.
16 = 2 4 
(2)4 = 16
Another fourth root of 16 is 2.

7. a a In fact, we have the following 4 cases for xn = a.
(i) When n is odd and a > 0:
a has only one positive nth root n a
e.g = 5, 3
53 = 125
which is positive.
(ii) When n is odd and a < 0:
a has only one negative nth root n a 3
e.g. 125 = 5,
(5)3 = 125
which is negative.

8. a a In fact, we have the following 4 cases for xn = a.
(iii) When n is even and a > 0:
a has two nth roots and  n a
e.g = 2, which is positive. 4 4
 16 = 2, which is negative.
24 = (2)4 = 16
(iv) When n is even and a < 0:
is not a real number. n a
e.g. 100 , 16 are not real numbers. 4

9. We can also use a calculator to find the values of radicals.
Follow-up question
Find the value of 3
216
We can also use a calculator to find the values of radicals.
216 6 =  ∵
216 6 3 = 6
216 3 = ∴

10. Follow-up question 216 Find the value of . By keying in: 6 216 = ∴3
216
By keying in:
gives 6
216 3 = ∴

11. In junior forms, we have learnt the definitions and laws of integral indices.

12. Zero and negative integral indices
If a  0, then
(i) a0 = 1
(ii) an = 1 an
n is a positive integer.
Laws of integral indices
If m and n are integers and a, b  0, then
(ii) am  an = = am  n am an
(i) am  an = am + n
(iii) (am)n = am  n
(iv) (ab)n = anbn
(v) n b a = ø ö ç è æ

13. How about if the index is a rational number,
for example ? 3 2 1 , - a
The meaning of
Using the law: (am)n = am  n, we have ∵ 2 1 ) ( a  = a = 2 1 a =
∴ is the square root of a. 2 1 a

14. a > 0 and n is a positive integer.∵ 3 1 ) ( a  = 3 1 a = a =
∴ is the cube root of a. 3 1 a
By similar argument, ∵
∴ is the nth root of a.
a > 0 and n is a positive integer.

15. where a > 0, m is an integer and n is a positive integer.
The meaning of
By the laws of indices and the above result,
and
Hence, we have
where a > 0, m is an integer and n is a positive integer.
For example,

16. Zero and negative rational indices
Now, we can extend the definitions and laws of indices to rational indices p and q.
Zero and negative rational indices
If a > 0, then
p is a rational number.
(i) a0 = 1
(ii) ap = 1 ap

17. Laws of rational indices
For rational numbers p and q, and positive numbers a and b:
(ii) ap  aq = = ap  q ap aq
(i) ap  aq = ap + q
(iii) (ap)q = ap  q
(iv) (ab)p = apbp
(v) p b a = ø ö ç è æ

18. Try to find the value of
Alternative Solution

19. value of with a calculator.
We can also find the
value of with a calculator.
By keying in:
gives ∴

20. Follow-up question
Find the value of
Alternative Solution

21. Follow-up question Find the value of . Using calculator, by keying in:
gives ∴