Honors Chapter 5 Project
Name _______________
SIMPLIFYING RADICAL EXPRESSIONS
25 , read “the square root of 25,” means one of the two positive equal factors of 25.
25  5  5  5 ;
36  6  6  6 ;
a2  a  a  a ;
a6  a3  a3  a3 ;
100  10 2  10 ;
49  7 2  7
m16  m 8  m 8  m 8 ;
y10  y 5
a is called a radical, a is called the radicand.
General Rules for Radicals:
x  y  xy
xy  x  y
x

y
x
y
x 2  ( x )2  x
2  5  10
18  9  2
8

2
8
2
1592  ( 159 )2  159
a x  b x  ( a  b) x
2 34 3 6 3
Simplifying Square Roots:
Sometimes it is convenient to leave square roots in radical form instead of using a calculator to find
approximations (decimal values). Look for perfect squares (i.e., 4, 9, 16, 25, 36, 49, ...) as factors of the number
that is inside the radical sign (radicand) and take the square root of any perfect square factor. Multiply the root
of the perfect square times the reduced radical. When there is an existing value that multiplies the radical,
multiply any root(s) times that value.
Examples:
1.
20  4  5
= 4 5
= 2 5
2.
48  16  3
=
16  3
= 4 3
3.
a5  a  a  a  a  a
=
aa a
= a2 a
(note that one a comes “out” for each pair of a’s in
the radical)
Simplify Completely:
1.
4
2.
16
3.
 100
4.
a8
5.
6.
a 6 b10
7.
9a 2
8.
 81m 64
9.
49a 4 b12
10.
121x14 y 6
11.
18
12.
125
13.
72
14.
180
15.
a3
w12
16.
b7
17.
m9
18.
75 x 7 y 5
19.
27a11b 7
20.
32a 7 b 4
21.
9a 8
22.
45a 7
23.
36 x 2 y 6
24.
12 x 20 y
25.
 200
Bonus: 2 45  2 5 
Rationalizing the Denominator means eliminating all irrational numbers from the denominator. When
eliminating a radical from a denominator, it is important that you always multiply by one – otherwise you will
change the value of the fraction. Multiplying by 1 (the multiplicative identity element) does not change the
value of what is being multiplied.
Single Radical Term Denominators: If your fraction has a denominator with only one term that contains a
radical, build a fraction with the radical in both its numerator and denominator. This fraction will always have a
value of 1, since its numerator and denominator are identical. Then multiply the original fraction by the new
fraction that you build. The radical in the denominator will be gone in the product, and the product will have
the same value as the original fraction.
Example
Given :
Example
9
Given:
6
3
5
Build a 1 
5
5
3
5 3 5
Multiply :


5
5
5
Build a 1 =
6
6
Multiply and Simplify:
Rewrite the following expressions with rational denominators.
Be sure to completely simplify your answers.
6
2
4
10
3
9
48
5
5
125
4
8
4
3 2
5
3
3
8
2
8
Bonus:
5 3
10
9
6 9 6 3 6



6
2
6
6