7. Principal nth root
Where n=positive integer greater than 1
= real number
Value for Value for
= positive real
number b
Such that
=negative real number
b
Such that
14. Simplify:
Step 1
Look for Perfect Squares
(Try to use the largest
perfect square possible.)
Step 2
Simplify Perfect Squares
Step 3
Multiply the numbers
inside and outside the
radical separately.
48
3 16
43
4 3
16. 2
a a
2
x x
Any even power is a perfect square.
4 2
10 5
90 45
x x
x x
x x
The square root
exponent is half of
the original
exponent.
17. Odd powers
When you take the square root of an odd power, the result is always an even
power and one variable left inside the radical.
5 2
11 5
91 45
x x x
x x x
x x x
18. Simplifying using variables
When you simplify an even power of a variable and the result is an odd power,
use absolute value bars to make sure your answer is positive.
14 7
14 12 7 6
x x
x y x y
Even powers
do not need
absolute
value.
20. a ab b
You can only multiply radicals by other radicals
8 3
Both under the radical
CAN multiply
8 3
Not under the radical
CANNOT multiply
21. What is an “nth Root?”
Extends the concept of square roots.
For example:
A cube root of 8 is 2, since 23 = 8
A fourth root of 81 is 3, since 34 = 81
For integers n greater than 1, if bn = a then b is an nth root of a.
Written where n is the index of the radical.
22. Rational Exponents
nth roots can be written using rational exponents.
For example:
In general, for any integer n greater than 1.
23. Real nth Roots
If n is odd:
a has one real nth root
If n is even:
And a > 0, a has two real nth roots
And a = 0, a has one nth root, 0
And a < 0, a has no real nth roots
24. Finding nth Roots
Find the indicated real nth root(s) of a.
Example: n = 3, a = -125
n is odd, so there is one real cube root: (-5)3 = -125
We can write
