1. Algebra 1
Section 11.2

2. Simplifying Radicals
In mathematics, answers are generally expected to be stated in simplest form.
When a radical is in simplest form, its radicand contains no perfect power factors.

3. Product Property of Radicals
For x ≥ 0 and y ≥ 0,
xy = x • y . n

4. Example 1
a) =
2 • 2 • 2 • 3
= • 6 =

5. Example 1
b) =
3 • 5 • 5
= • 3 =

6. Example 1
c) = 3
2 • 2 • 2 • 5 3
= • 5 3 = 3

7. Simplifying Radicals
Write the radicand as the product of its prime factors.
Apply the Product Property to the perfect powers.

8. Simplifying Radicals Simplify the roots of the perfect powers.
Multiply the factors inside and outside the radical.

9. Example 2 a)
= 2 • 2 • 2 • 3 • 3 • 3 • 3
= • 2
32 •
= 2 • 3 • 3 • 2 =

10. Example 2 b) 14,580 = 2 • 2 • 3 • 3 • 3 • 3 • 3 • 3 • 5 = 33 •
= 2 • 2 • 3 • 3 • 3 • 3 • 3 • 3 • 5 3
= •
2 • 2 • 5
33 • 3
= 3 • 3 • 20 3 = 3

11. Even and Odd Roots
If the index is even (an “even root”) and the radical contains variables, care must be taken to use an absolute value:
For example, x2 = |x|

12. Even and Odd Roots
If the index is odd (an “odd root”), no absolute value is required.
For example, x3 = x

13. Definition The nth root of xn or xn = |x| if n is even. x if n is odd.

14. Example 3 Simplify 32x3y2 . = 16 • 2x x2 • y2 • = 4 • |x| • |y| • 2x
= • 2x
x2 •
y2 •
= 4 • |x| • |y| • 2x
= 4x|y| 2x

15. Example 4 Simplify 16x2y4z6 . = 8 • 2x2y3yz3z3 = 8y3z3z3 • 2x2y

16. Even and Odd Roots
The absolute value is not always required for even roots.
It is necessary only when the simplification of an even root of a variable raised to an even power produces a variable with an odd exponent.

17. Even and Odd Roots x2 = |x| x5 = x2 x x3 = x x x6 = |x|3 x4 = x2

18. Homework: pp