Operations with Radicals
Example 1
Simplify Expressions
Simplify.
a. 50a5 x4
5 4
2
2 2
2
= 2 • 5 •
= 5a2x2 2a
b.
d
2 2
)
•
2 2
a •
(x )
Product Property of Radicals
Simplify.
c.
9
9
9
3
2m
12
n
(a
Factor into squares where possible.
12
n
d
2 2
2 • 5 • (a ) • a • (x )
=
50a x
d
=
12
n
9
2m
(d 6 )2
=
(d 6 )2
(n )
=
=
•
n
4
d6
n
d
n
6
n
Example 2
=d ,
(d )
n
4
6
6 2
4
=
4
2
(n ) = n
•
n
3
n
n
n
5
Quotient
Property
9
2m
•
3
4m 2
3
2
4m
9 • 4m 2
3
2 m • 4m
3
36 m2
3
3
=
Rationalize the denominator.
9
2m
3
=
Product Property
d6
n
3
3
(n ) • n
4 2
3
=
=
Factor into squares.
4 2
=
=
9
3
Quotient Property
8m 3
36 m2
2m
2
Rationalize the
denominator.
Product
Property
Multiply.
3
8m3 = 2m
n=n
Multiply Radicals
Simplify 2 18a 3 • 5 72a 2b3 .
3
2 3
3
Product Property of Radicals
2 3
2 18a • 5 72a b = 2 • 5 • 18 a • 72a b
= 10 •
(36)2 • (a 2 )2 • a • b2 • b
= 10 •
36
2
•
2
(a 2 ) 2 • a • b • b
= 10 • 36 • a2 • b •
a •
b or 360a2b
ab
Factor into squares where
possible.
Product Property of Radicals
Multiply.
Example 3
Add and Subtract Radicals
Simplify –4 18 + 5 50 – 2 98
–4 18 + 5 50 – 2 98
2
2
= -4 2 • 3 + 5 2 • 5 – 2 2 • 7
2
Factor using squares.
2
2
= -4 2 • 3 + 5 2 • 5 – 2 2 • 7
= -4 • 3 • 2 + 5 • 5 • 2 – 2 • 7 • 2
Product Property
2
32 = 3,
= -12 2 + 25 2 – 14 2
=- 2
Example 4
Simplify.
a. (2 2 –
52 = 5,
72 = 7
Multiply.
Combine like radicals.
Multiply Radicals
7 )(3 7 – 5)
F
O
I
L
7 )(3 7 – 5) = 2 2 • 3 7 + 2 2 • (–5) + (– 7 ) • (3 7 ) + (– 7 ) • (–5)
(2 2 –
2
= 6 14 – 10 2 – 3 7 + 5 7
Product Property
= 6 14 – 10 2 – 21 + 5 7
3 7
2
= 3 7 = 21
b. (4 5 – 3 2 )(4 5 + 3 2 )
(4 5 – 3 2 )(4 5 + 3 2 )
= 4 5 • 4 5 + 4 5 ( 3 2 ) + (– 3 2 )(4 5 ) + (– 3 2 )( 3 2 )
2
= 16 5 + 12 10 – 12 10 – 9
= 80 – 18
2
Multiply.
2
52 = 5,
= 62
7
2
5
22 = 2
Subtract.
Example 5
Simplify
FOIL
Use a Conjugate to Rationalize a Denominator
7
2+ 5
=
=
7
(2
=
2
•
5)
2 7 –
2
=
.
(2 –
(2 –
35
2
–
5)
5)
(2 – 5)
because 2 – 5 is the conjugate of 2 +
(2 – 5)
Distributive Property for the numerator
FOIL for the denominator
Multiply by
(5)
2 7 –
35
4 – 5
2 7 – 35
–1
= –2 7 +
35
Multiply.
Combine like terms.
Multiply numerator and denominator by –1.
5.