1. Algebra 1
Section 11.4

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

3. Quotient Property of Radicals
For x ≥ 0 and y > 0, x n y =

4. Example 1
Simplify ÷ 40 5 = 40 5 = 8 =
2 2

5. Example 2
Simplify: 15 12 5 4 = = 15 12 = 5 4 5 2 =

6. Simplifying Radicals
Fractions with radicals in the denominator or radicals with fractions in the radicand are not considered to be simplified.

7. Definition
The process of removing radicals from the denominator of a fraction is called rationalizing the denominator.

8. Example 3
Simplify: 5 7 7 • = 35 7

9. Example 4
Simplify: 3 6 1 2 = = 3 6 = 1 2 1 2 2 • = 2

10. Example 5
Simplify: 2 75 2
5 3 = 3 • = 6
5 • 3 6 15 =

11. Rationalizing the Denominator
Write both radicals in fractional form.
Simplify both the numerator and the denominator as much as possible.

12. Rationalizing the Denominator
If a radical is left in the denominator, multiply by the form of one that makes the radicand in the denominator a perfect power.

13. Rationalizing the Denominator
Simplify both the numerator and the denominator and reduce the fraction if necessary.

14. Example 7 Simplify: 8 16 1 2 = = 8 16 = 1 2 22 • = 4 2 1 2 3 3 3 3 3 3

15. Example 8 Simplify: 18a2bc2 9ab3c3 = 18a2bc2 9ab3c3 = 2a b2c 2a b c c• =
2ac bc

16. Homework: pp