6-1 Roots and Radical Expressions
Quick Review
You can simplify a radical expression by finding the roots. The principal root of a number with two real roots is the positive root. The principal
nth root of b is written as
b
n
,
the th , root of b , , comma where b is the radicand and n is the index of the radical expression.
For any real number a,
a
n
n
=
{
a
if
n
is odd
|
a
|
if
n
is even
.
the th , root of eh to the n end root , , equals . left brace . table with 2 rows and 1 column , row1 column 1 , eh , if , n , isodd , row2 column 1 , absolute value of eh , . if , n , iseven , end table . .
Example
What is the simplest form of
36
x
6
?
square root of 36 , x to the sixth end root . question mark
6
2
x
6
Find the root of the integer
.
=
6
2
(
x
3
)
2
Find the root of the variable
.
=
6
|
x
3
|
Take the square root of each term
. Since the index
is even, include the absolute value symbol to ensure that
the root is positive even when
x
3
is negative
.
table with 3 rows and 2 columns , row1 column 1 , square root of 6 squared , x to the sixth end root , column 2 cap findtherootoftheinteger . . , row2 column 1 , equals . square root of 6 squared . open , x cubed , close squared end root , column 2 cap findtherootofthevariable . . , row3 column 1 , equals 6 absolute value of , x cubed , end absolute value , , column 2 table with 3 rows and 1 column , row1 column 1 , cap takethesquarerootofeachterm . .cap sincetheindex , row2 column 1 , isevencommaincludetheabsolutevaluesymboltoensurethat , row3 column 1 , therootispositiveevenwhen . x cubed . isnegative . . , end table , end table
Exercises
Find each real root.
-
25
square root of 25
-
0.49
square root of 0.49
-
−
8
3
cube root of negative 8 end root ,
-
−
8
3
negative , cube root of 8 ,
Simplify each radical expression. Use absolute value symbols when needed.
-
81
x
2
square root of 81 , x squared end root
-
64
x
6
3
cube root of 64 , x to the sixth end root ,
-
16
x
12
4
the fourth , root of 16 , x to the twelfth end root ,
-
0.00032
x
5
5
the fifth , root of 0.00032 . x to the fifth end root ,
-
9
x
4
36
square root of fraction 9 , x to the fourth , over 36 end fraction end root
-
125
x
6
y
9
3
cube root of 125 , x to the sixth , y to the ninth end root ,
6-2 Multiplying and Dividing Radical Expressions
Quick Review
If
a
n
the th , root of eh , and
b
n
the th , root of b , are real numbers, then
(
a
n
)
(
b
n
)
=
a
b
n
,
open , the th , root of eh , , close . open , the th , root of b , , close . equals , the th , root of eh b end root , , comma and if
b
≠
0
,
b not equal to 0 comma then
a
n
b
n
=
a
b
n
.
fraction the th , root of eh , , over the th , root of b , end fraction . equals , the th , root of eh over b end root , , .
To rationalize the denominator of an expression, rewrite it so that the denominator contains no radical expressions.
Example
What is the simplest form of
32
x
2
y
⋅
18
x
y
3
?
square root of 32 , x squared , y end root . dot . square root of 18 x , y cubed end root . question mark
(
32
x
2
y
)
(
18
x
y
3
)
Combine terms
.
=
(
4
2
⋅
2
x
2
y
)
(
3
2
⋅
2
x
y
3
)
Factor
.
=
4
2
⋅
3
2
⋅
2
2
x
3
y
4
Consolidate like terms
.
=
4
2
⋅
3
2
⋅
2
2
(
x
2
x
)
(
y
2
)
2
Identify perfect squares
.
=
4
⋅
3
⋅
2
x
y
2
x
=
24
x
y
2
x
Extract perfect squares
.
table with 5 rows and 2 columns , row1 column 1 , square root of open . 32 , x squared , y . close . open . 18 x , y cubed . close end root , column 2 cap combineterms . . , row2 column 1 , equals . square root of open . 4 squared , dot 2 , x squared , y . close . open . 3 squared , dot 2 x , y cubed . close end root , column 2 cap factor , . , row3 column 1 , equals . square root of 4 squared , dot , 3 squared , dot , 2 squared , x cubed , y to the fourth end root , column 2 cap consolidateliketerms . . , row4 column 1 , equals . square root of 4 squared , dot , 3 squared , dot , 2 squared . open , x squared , x , close . open , y squared , close squared end root , column 2 cap identifyperfectsquares . . , row5 column 1 , equals 4 dot 3 dot 2 x , y squared , square root of x equals 24 x , y squared , square root of x , column 2 cap extractperfectsquares . . , end table
Exercises
Multiply if possible. Then simplify.
-
9
3
⋅
3
3
cube root of 9 , , dot , cube root of 3 ,
-
−
7
3
⋅
49
3
cube root of negative 7 end root , , dot , cube root of 49 ,
-
2
⋅
8
square root of 2 dot square root of 8
Multiply and simplify.
-
8
x
2
⋅
2
x
2
square root of 8 , x squared end root , dot , square root of 2 , x squared end root
-
5
9
y
2
3
⋅
24
y
3
5 . cube root of 9 , y squared end root , . dot , cube root of 24 y end root ,
Divide and simplify.
-
128
8
square root of 128 over 8 end root
-
81
x
5
y
3
3
3
x
2
3
fraction cube root of 81 , x to the fifth , y cubed end root , , over cube root of 3 , x squared end root , end fraction
-
162
x
4
4
2
y
8
4
fraction the fourth , root of 162 , x to the fourth end root , , over the fourth , root of 2 , y to the eighth end root , end fraction
Divide. Rationalize all denominators.
-
8
6
fraction square root of 8 , over square root of 6 end fraction
-
3
x
5
8
x
2
fraction square root of 3 , x to the fifth end root , over 8 , x squared end fraction
-
6
x
2
y
4
3
2
5
x
7
y
3
fraction cube root of 6 , x squared , y to the fourth end root , , over 2 . cube root of 5 , x to the seventh , y end root , end fraction
