Download Lecture Notes for Section 8.2

Elementary Algebra Notes
Section 8.2
Page 1 of 4
Section 8.2: Multiplying, Dividing, and Simplifying Radicals
Big Idea: There are rules for how to multiply and divide expressions with radicals which can lead to simpler
radical expressions.
Big Skill: You should be able to simplify radicals by switching them between products or quotients of radicals
and radicals of products or quotients.
Product Rule for Square Roots: For nonnegative real numbers a and b,

a  b  ab (the product of two square roots is the square root of the products)
 and ab  a  b (vice-versa; the square root of a product equals the product of the square roots).
This second equation is very useful for simplifying radicals, if you can think of the number under the square
root as a product of a perfect square and another number.
Criteria for a Simplified Radical Expression:

There are as few radicals in the expression as possible.

The radicands are as small as possible.
Practice: Simplify the following radicals as much as possible.
1.
6  11 
2.
13  x 
3.
10  10 
4.
20 
5.
60 
6.
500 
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra Notes
7.
17 
8.
10  50 
9.
6 2 
Section 8.2
Page 2 of 4
Quotient Rule for Square Roots: For nonnegative real numbers a and b, and b  0,
a
a

(the square root of a quotient is the quotient of the square roots)

b
b

And
a
a
(the quotient of two square roots is the square root of the quotient)

b
b
Practice: Simplify the following radicals as much as possible.
4
10.

49
11.
48

3
12.
5

6
13.
8 50

4 5
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra Notes
Section 8.2
Page 3 of 4
3 7


8 2
14.
The Square Root of a Square: For any number a,


a 2  a (the square root of a square is the absolute value of squared value)
Notice:
 4
2
 16  4  4
Practice: Simplify the following radicals as much as possible.
x4 
15.
16. 100 p8 
7

y4
17.
Product Rule for Radicals: For all real numbers for which the indicated roots exist,
 n a  n b  n ab

n
n
a na

b
b
Practice: Simplify the following radicals as much as possible.
18. 3 108 
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.
Elementary Algebra Notes
Section 8.2
Page 4 of 4
19. 4 160 
20.
4
16

625
21.
3
z9 
22.
3
8x6 
23.
3
54t 5 
24.
3
a15

64
Algebra is:
the study of how to perform multi-step arithmetic calculations more efficiently,
and the study of how to find the correct number to put into a multi-step calculation to get a desired answer.