Roots and Powers
The Relationship Between Roots and Powers
Letâ€™s look again at the relationship between taking a square root
and squaring.
For example, letâ€™s start with the number 6.

6 

Take its square root.



Square the square root.



Write the product as a single radical.



Multiply.



Simplify.


= 6 
The result, 6, is the number that we started with. 


Thus, squaring â€œundoesâ€ taking a square root.
Here, we saw that
= 6.
This is true in general.
Property â€”
Raising an nth Root to the n^{th} Power
English Raising the nth root of a number to the n^{th} power results in
the original number.
Algebra If
is defined, then
Here, n is a positive integer.
Example
Example 1
Simplify:
Solution
Use this relationship to simplify each radical:
Note:
only when x
≥ 0. An even root
of a negative number is not a real number.
Now, letâ€™s see what happens when we first square a negative number, and
then take its square root.
For example, letâ€™s start with the number 4.
Square the number. The result is positive.

4 (4)^{2} 
= 16 
Take the square root.



Simplify. 

= 4 
The result, 4, is the opposite of the number we started with.
Thus, when we square a negative number, and then take the square root,
we obtain the opposite of the original number.
This relationship holds true, in general for even roots.
