Section 7.5
Multiplying
With More
Than One Term
and
Rationalizing
Denominators
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Objective #1
Multiply radical expressions with more
than one term.
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Multiplying
Radical expressions with more than one term are
multiplied in much the same way that polynomials with
more than one term are multiplied. That is, using the
distributive property and FOIL method.
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Multiplying Radicals
EXAMPLE
(
Multiply: (a) 3 3 3 6 + 7 3 4
SOLUTION
(
(a) 3 3 3 6 + 7 3 4
(b)
(
)
(b)
(
3− 2
)( 10 − 11).
)
= 3 3 ⋅3 6 + 3 3 ⋅7 3 4
Use the distributive property.
= 3 18 + 7 3 12
Multiply the radicals.
)( 10 − 11)
3 ⋅ 10 + 3 (− 11 )+ (− 2 ) 10 + (− 2 )(− 11 )
3− 2
=
= 30 − 33 − 20 + 22
Use FOIL.
Multiply the radicals.
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Multiplying Radicals
CONTINUED
= 30 − 33 − 4 ⋅ 5 + 22
Factor the third radicand using
the greatest perfect square factor.
= 30 − 33 − 4 ⋅ 5 + 22
Factor the third radicand into
two radicals.
= 30 − 33 − 2 5 + 22
Simplify.
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Objective #1: Example
1a. Multiply:
(
(
6 x + 10
)
6 x + 10 =
)
6 ⋅ x + 6 10
= x 6 + 60
= x 6 + 4 ⋅15
= x 6 + 2 15
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Objective #1: Example
1a. Multiply:
(
(
6 x + 10
)
6 x + 10 =
)
6 ⋅ x + 6 10
= x 6 + 60
= x 6 + 4 ⋅15
= x 6 + 2 15
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Objective #1: Example
(
)(
1b. Multiply: 6 5 + 3 2 2 5 − 4 2
(6
)(
5+3 2 2 5−4 2
)
)
= (6 5)(2 5) + (6 5)( −4 2) + (3 2)(2 5) + (3 2)( −4 2)
= 12 ⋅ 5 − 24 10 + 6 10 − 12 ⋅ 2
60 − 24 10 + 6 10 − 24
=
= 36 − 18 10
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Objective #1: Example
(
)(
1b. Multiply: 6 5 + 3 2 2 5 − 4 2
(6
)(
5+3 2 2 5−4 2
)
)
= (6 5)(2 5) + (6 5)( −4 2) + (3 2)(2 5) + (3 2)( −4 2)
= 12 ⋅ 5 − 24 10 + 6 10 − 12 ⋅ 2
60 − 24 10 + 6 10 − 24
=
= 36 − 18 10
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Objective #2
Use polynomial special products to multiply
radicals.
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Special Products
Some radicals can be multiplied using the special
products for multiplying polynomials.
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Objective #2: Example
2a. Multiply:
(
(
5+ 6
) ( 5)
2
5 + 6=
2
)
2
+ 2⋅ 5⋅ 6 +
( 6)
2
5 + 2 30 + 6
=
= 11 + 2 30
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Objective #2: Example
2a. Multiply:
(
(
5+ 6
) ( 5)
2
5 + 6=
2
)
2
+ 2⋅ 5⋅ 6 +
( 6)
2
5 + 2 30 + 6
=
= 11 + 2 30
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Objective #2: Example
2b. Multiply:
(
a− 7
(
)(
a− 7
)(
a+ 7
)
) ( a) − ( 7)
a+ 7=
2
2
= a−7
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Objective #2: Example
2b. Multiply:
(
a− 7
(
)(
a− 7
)(
a+ 7
)
) ( a) − ( 7)
a+ 7=
2
2
= a−7
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Objective #3
Rationalize denominators containing one term.
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Rationalizing the Denominator
Rationalizing the denominator is a process that
involves rewriting a radical expression as an equivalent
expression in which the denominator no longer contains
any radicals.
When the denominator contains a single radical with an
nth root, multiply the numerator and denominator by
a radical of index n that produces a perfect nth power
in the denominator’s radicand.
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Rationalizing Denominators
EXAMPLE
Rationalize each denominator: (a) 3
5
y2
(b)
10
5
16 x
2
.
SOLUTION
3
5
5
3
as
(a) Using the quotient rule, we can express y 2
2 . We
3
y
have cube roots, so we want the denominator’s radicand to be a
2
perfect cube. Right now, the denominator’s radicand is y . We
know that 3 y 3 = y. If we multiply the numerator and the
3
5
by 3 y , the denominator becomes
denominator of
3
y2
3
y 2 ⋅ 3 y = 3 y 3 = y.
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Rationalizing Denominators
CONTINUED
The denominator no longer contains a radical. Therefore, we
3 y
multiply by 1, choosing
for 1.
3 y
3
5
=
2
y
=
=
3
Use the quotient rule and rewrite
as the quotient of radicals.
5
y2
3
3
5
2
3
y
3
5y
3
3
y
⋅
3
y
3
y
Multiply the numerator and
denominator by 3 y to remove
the radical in the denominator.
Multiply numerators and
denominators.
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Rationalizing Denominators
CONTINUED
5y
=
y
3
Simplify.
(b) The denominator, 5 16x 2 is a fifth root. So we want the
denominator’s radicand to be a perfect fifth power. Right now,
4 2
2
the denominator’s radicand is 16 x or 2 x . We know that
5
25 x 5 = 2 x. If we multiply the numerator and the denominator
5
10
2 x3
of
, the denominator becomes
by
2
3
5
5
16 x
2x
5
16 x 2 ⋅ 5 2 x 3 = 5 2 4 x 2 ⋅ 5 2 x 3 = 5 25 x 5 = 2 x.
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Rationalizing Denominators
CONTINUED
The denominator’s radicand is a perfect 5th power. The
denominator no longer contains a radical. Therefore, we
5
2 x3
multiply by 1, choosing
for 1.
3
5
2x
10
5
16 x 2
=
=
=
10
Write the denominator’s radicand
as an exponential expression.
24 x 2
5
10
4
5
2 x
2
⋅
10 5 2 x 3
5
25 x 5
5
2 x3
5
2 x3
Multiply the numerator and the
denominator by 5 2 x 3 .
Multiply the numerators and
denominators.
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Rationalizing Denominators
CONTINUED
105 2 x 3
=
2x
Simplify.
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Objective #3: Example
3a. Rationalize the denominator:
3
=
7
3
7
3 7
⋅
7 7
=
21
49
21
=
7
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Objective #3: Example
3a. Rationalize the denominator:
3
=
7
3
7
3 7
⋅
7 7
=
21
49
21
=
7
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Objective #3: Example
3b. Rationalize the denominator:
6x
5
8x2 y 4
=
6x
5
8x2 y 4
6x
5 3 2 4
2 x y
6x
=
⋅
5 2 3
2 x y
5 3 2 4 5 2 3
2 x y
=
2 x y
6 x 5 22 x3 y
5 5 5 5
2 x y
6 x 5 22 x3 y
=
2 xy
35 4 x3 y
=
y
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Objective #3: Example
3b. Rationalize the denominator:
6x
5
8x2 y 4
=
6x
5
8x2 y 4
6x
5 3 2 4
2 x y
6x
=
⋅
5 2 3
2 x y
5 3 2 4 5 2 3
2 x y
=
2 x y
6 x 5 22 x3 y
5 5 5 5
2 x y
6 x 5 22 x3 y
=
2 xy
35 4 x3 y
=
y
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Objective #4
Rationalize denominators containing two terms.
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Rationalizing the Denominator
Rationalize the denominator when the denominator
contains two terms with one or more square root,
multiply the numerator and denominator by the
conjugate of the denominator.
The product of the denominator and its conjugate is
found using the formula
( A + B)( A − B) = A2 − B 2 .
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Rationalizing Denominators
EXAMPLE
12
Rationalize each denominator: (a)
7+ 3
(b)
3 x+ y
y −3 x
.
SOLUTION
(a) The conjugate of the denominator is 7 − 3. If we
multiply the numerator and the denominator by 7 − 3 , the
simplified denominator will not contain a radical. Therefore, we
7− 3
multiply by 1, choosing
for 1.
7− 3
12
12
7− 3
=
⋅
7+ 3
7+ 3 7− 3
Multiply by 1.
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Rationalizing Denominators
CONTINUED
12
12
7− 3
=
⋅
7+ 3
7+ 3 7− 3
12 7 − 3
=
2
2
7 − 3
(
)
( ) ( )
12( 7 − 3 )
=
(
7−3
12 7 − 3
=
4
3
(
12 7 − 3
=
41
)
)
Multiply by 1.
( A + B )( A − B ) = A2 − B 2
Evaluate the exponents.
Subtract.
Divide the numerator and
denominator by 4.
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Rationalizing Denominators
CONTINUED
(
)
= 3 7 − 3 or 3 7 − 3 3
Simplify.
(b) The conjugate of the denominator is y + 3 x . If we
multiply the numerator and the denominator by y + 3 x , the
simplified denominator will not contain a radical. Therefore, we
y +3 x
multiply by 1, choosing
for 1.
y +3 x
3 x+ y
y −3 x
=
3 x+ y
y −3 x
⋅
y +3 x
y +3 x
Multiply by 1.
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Rationalizing Denominators
CONTINUED
3 x+ y
y −3 x
=
3 x+ y
y −3 x
y +3 x
⋅
y +3 x
3 x+ y 3 x+ y
=
⋅
y −3 x
y +3 x
Multiply by 1.
Rearrange terms in the second
numerator.
(
3 x ) + 2 ⋅ (3 x )( y ) + ( y )
=
( y ) − (3 x )
2
2
2
9 x + 6 xy + y
=
y − 9x
2
( A + B )2 = A2 + 2 AB + B 2
( A + B )( A − B ) = A2 − B 2
Simplify.
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Objective #4: Example
18
4. Rationalize the denominator:
2 3+3
18
18
2 3−3
=
⋅
2 3+3 2 3+3 2 3−3
36 3 − 54 36 3 − 54
= =
12 − 9
22 ⋅ 3 − 32
36 3 − 54 3(12 3 − 18)
= =
3
3
= 12 3 − 18
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Objective #4: Example
18
4. Rationalize the denominator:
2 3+3
18
18
2 3−3
=
⋅
2 3+3 2 3+3 2 3−3
36 3 − 54 36 3 − 54
= =
12 − 9
22 ⋅ 3 − 32
36 3 − 54 3(12 3 − 18)
= =
3
3
= 12 3 − 18
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Objective #5
Rationalize numerators.
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Rationalizing the Numerator
Rationalizing the numerator is a process that involves
rewriting a radical expression as an equivalent expression
in which the numerator no longer contains any radicals.
To rationalize the numerator when the numerator
contains a radical, multiply by 1, using the same methods
we used to rationalize the denominator.
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Rationalizing Numerators
EXAMPLE
Rationalize the numerator:
x+7 − x
.
7
SOLUTION
The conjugate of the numerator is x + 7 + x . If we multiply
the numerator and the denominator by x + 7 + x , the
simplified numerator will not contain a radical. Therefore, we
x+7 + x
multiply by 1, choosing
for 1.
x+7 + x
x+7 − x
=
7
x+7 − x x+7 + x
⋅
7
x+7 + x
Multiply by 1.
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Rationalizing Numerators
CONTINUED
=
(
) ( )
(
)
2
x+7 − x
7 x+7 + x
x+7−x
=
7 x+7 + x
7
=
7 x+7 + x
=
2
( A + B )( A − B ) = A2 − B 2
Leave the denominator in
factored form.
(
)
Evaluate the exponents.
(
)
Simplify the numerator.
1
x+7 + x
Simplify by dividing the
numerator and denominator
by 7.
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Objective #5: Example
x+3− x
3
5. Rationalize the numerator:
x+3− x
=
3
x+3− x x+3+ x
⋅
3
x+3+ x
x + 3) − ( x )
(=
2
=
=
=
2
3 x+3+3 x
3
3
(
x+3+ x
x+3− x
3 x+3+3 x
)
1
x+3+ x
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Objective #5: Example
x+3− x
3
5. Rationalize the numerator:
x+3− x
=
3
x+3− x x+3+ x
⋅
3
x+3+ x
x + 3) − ( x )
(=
2
=
=
=
2
3 x+3+3 x
3
3
(
x+3+ x
x+3− x
3 x+3+3 x
)
1
x+3+ x
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