1. Indices

2. “nth” nth Root of a Real Number
The "nth Root" used n times in a multiplication gives the original value
" nth ? "
1st, 
2nd, 
3rd, 
4th, 
5th, 
…, 
nth,  … 
Instead of talking about the "4th", "16th", etc, if we want to talk generally we say the
“nth”

3. Just like the square root is
The nth Root 2
Just like the square root is
used two times in a multiplication to get the original value.

4. used three times in a multiplication to get
The nth Root 3
The cube root is
used three times in a multiplication to get
the original value.

5. used n times in a multiplication to get the original value.
The nth Root n
The nth root is
used n times in a multiplication to get the original value.

6. .
Let a be a real number and n be a positive integer. If there exists a real number r such that
rn = a
Then r is called the nth root of a. The nth root which has the same sign
as a is denoted by

7. The nth Root Symbol
This is the special symbol that means "nth root", it is the "radical" symbol (used for square roots) with a little n to mean nth root.

8. 42 = 16
is the second root or
square root of 4 16

9. (-42) = 16 -4
is also a square root of 16
It is denote by

10. -7 and 5 are the third roots or cube roots of
53 = 125
(-7)3 = -343
-7 and 5 are the third roots or cube roots of
-343 and 125 respectively.

11. -3 and 3 are the fourth roots of 81.34
= (-3)4
= 81
-3 and 3 are the
fourth roots of 81.
Positive 4th root
Negative 4th root

12. Positive nth root Negative nth root
when a  0 and n is an even integer.
There are two real nth roots of a.
Positive nth root
Negative nth root

13. that when a  0 and n is even,
Note
that when a  0 and n is even,
is undefined in the real number system. Thus,
undefined

14. a = 0 a  0 a  0, n is odd number Conclusion of nth roots of a or is
positive number
a  0
is negative number
a  0,
n is odd number

15. Ex1
Find the values of the following.
Write 64 in the form r3

16. Write 625 in the form r4 is also a 4th root of 625, but
only present the positive 4th root.

18. Write -8 in the form r3