Algebra Review: Simplifying Radicals
Use With Lesson 6-7
A radical expression is in simplest form when all of the following are true.
- The radicand has no perfect square factors other than 1.
- The radicand does not contain a fraction.
- A denominator does not contain a radical expression.
Example 1
Simplify the expressions
2
⋅
8
square root of 2 dot square root of 8 and
294
÷
3
.
square root of 294 , divides square root of 3 .
|
2
⋅
8
=
2
⋅
8
Write both numbers under one radical.
=
16
Simplify the expression under the radical.
=
4
Factor out perfect squares and simplify.
table with 3 rows and 3 columns , row1 column 1 , square root of 2 dot square root of 8 , column 2 equals , square root of 2 dot 8 end root , column 3 cap write both numbers under one radical. , row2 column 1 , , column 2 equals square root of 16 , column 3 cap simplify the expression under the radical. , row3 column 1 , , column 2 equals 4 , column 3 cap factor out perfect squares and simplify. , end table
|
294
÷
3
=
294
3
=
98
=
49
⋅
2
=
7
2
table with 4 rows and 2 columns , row1 column 1 , square root of 294 , divides square root of 3 , column 2 equals , square root of 294 over 3 end root , row2 column 1 , , column 2 equals square root of 98 , row3 column 1 , , column 2 equals , square root of 49 dot 2 end root , row4 column 1 , , column 2 equals 7 square root of 2 , end table
|
Example 2
Write
4
3
square root of 4 thirds end root
in simplest form.
4
3
=
4
3
Rewrite the single radical as a quotient.
=
2
3
Simplify the numerator.
=
2
3
⋅
3
3
Multiply by
3
3
(
a form of
1
)
to remove the radical from the denominator.
This is called
rationalizing the denominator.
=
2
3
3
table with 4 rows and 3 columns , row1 column 1 , square root of 4 thirds end root , column 2 equals , fraction square root of 4 , over square root of 3 end fraction , column 3 cap rewrite the single radical as a quotient. , row2 column 1 , , column 2 equals , fraction 2 , over square root of 3 end fraction , column 3 cap simplify the numerator. , row3 column 1 , , column 2 equals , fraction 2 , over square root of 3 end fraction , dot , fraction square root of 3 , over square root of 3 end fraction , column 3 table with 2 rows and 1 column , row1 column 1 , cap multiply by . fraction square root of 3 , over square root of 3 end fraction , open . a form of . 1 close . to remove the radical from the denominator. , row2 column 1 , cap this is called . rationalizing the denominator. , end table , row4 column 1 , , column 2 equals , fraction 2 square root of 3 , over 3 end fraction , column 3 , end table
Exercises
Simplify each expression.
-
5
⋅
10
square root of 5 dot square root of 10
-
243
square root of 243
-
128
÷
2
square root of 128 , divides square root of 2
-
125
4
square root of 125 over 4 end root
-
6
⋅
8
square root of 6 dot square root of 8
-
36
3
fraction square root of 36 , over square root of 3 end fraction
-
144
2
fraction square root of 144 , over square root of 2 end fraction
-
3
⋅
12
square root of 3 dot square root of 12
-
72
÷
2
square root of 72 divides square root of 2
-
169
square root of 169
-
28
÷
8
28 divides square root of 8
-
300
÷
5
square root of 300 , divides square root of 5
-
12
⋅
2
square root of 12 dot square root of 2
-
6
⋅
3
9
fraction square root of 6 dot square root of 3 , over square root of 9 end fraction
-
3
⋅
15
2
fraction square root of 3 dot square root of 15 , over square root of 2 end fraction
