Transcript Slide 1

§ 7.5
Multiplying With More Than One Term and
Rationalizing Denominators
Section objectives
In this section, you will learn to:
Multiply radical expressions having more than one term
Use polynomial special products to multiply radicals
Rationalize the denominators containing one term
Rationalize the denominators containing two terms
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.5
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.
 3  2  10  11
 3  10  3 11  2  10   2  11
 30  33  20  22
Use FOIL.
Multiply the radicals.
Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.5
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.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.5
Multiplying Radicals
EXAMPLE
Multiply: (a)
SOLUTION
(a)

3

x 3
3

x 7
3

x 3
3
x 7

(b)


2
2x  y .

 3 x  3 x  3 x  7   33 x   37
Use FOIL.
 3 x2  73 x  33 x  21
Multiply the radicals.
 3 x 2  73 x  33 x  21
Group like terms.
 3 x2  43 x  21
Combine radicals.


Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.5
Multiplying Radicals
CONTINUED
(b)


2x  y

2
 2x   2 
2
2x  y 
 2x  2 2xy  y
 y
Use the special product
for  A  B 2 .
2
Multiply the radicals.
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.5
Rationalizing Denominators
EXAMPLE
5
Rationalize each denominator: (a) 3 2
y
(b)
SOLUTION
(a) Using the quotient rule, we can express
3
10
5
16x
2
5
as
2
y
.
3
3
5
y
2
. We
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
denominator of
3
5
y
2
3
by
3
y , the denominator becomes
y 2  3 y  3 y 3  y.
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.5
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.
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.5
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,
2
4 2
the denominator’s radicand is 16x or 2 x . We know that
5
25 x5  2x. If we multiply the numerator and the denominator
of
10
5
16 x
2
5
by
5
2 x3
5
3
2x
, the denominator becomes
16x2  5 2x3  5 24 x2  5 2x3  5 25 x5  2x.
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.5
Rationalizing Denominators
CONTINUED
The denominator’s radicand is a perfect 5th power. The
denominator no longer contains a radical. Therefore, we
multiply by 1, choosing
10
5
16x 2



10
10
4
5
2 x
2

10 5 2 x 3
5
25 x 5
2 x3
5
3
2x
for 1.
Write the denominator’s radicand
as an exponential expression.
24 x 2
5
5
5
2 x3
5
2 x3
Multiply the numerator and the
denominator by 5 2x3 .
Multiply the numerators and
denominators.
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.5
Rationalizing Denominators
CONTINUED
105 2 x 3

2x
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.5
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.
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.5
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  B2
Evaluate the exponents.
Subtract.
Divide the numerator and
denominator by 4.
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.5
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
multiply by 1, choosing
3 x y
y 3 x

3 x y
y 3 x

y 3 x
y 3 x
y 3 x
y 3 x
for 1.
Multiply by 1.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.5
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  B2
 A  BA  B  A2  B2
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.5
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
x7  x x7  x


7
7
x7  x
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.5
Multiply by 1.
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  B2
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.
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.5
In Summary…
Important to Remember:
Radical expressions that involve the sum and difference of the same
two terms are called conjugates. To multiply conjugates, use
( A  B)( A  B)  A2  B 2
The process of rewriting a radical expression as an equivalent
expression without any radicals in the denominator is called
rationalizing the denominator.
GET THOSE RADICALS OUT OF THE DENOMINATOR!!!!
Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.5
In Summary…
On Rationalizing the Denominator…
If the denominator is a single radical term with nth root:
See what expression you would need to multiply by to obtain a perfect nth power in the
denominator. Multiply numerator and denominator by that expression.
If the denominator contains two terms:
Rationalize the denominator by multiplying the numerator and the denominator by
the conjugate of the denominator.
More than two terms in the denominator and rationalizing can get very complicated. Note that you
don’t have rules here for those situations. To rationalize simply means to “get the radical out”. By
common agreement, we usually rationalize the denominator in a rational expression. We make the
denominator “nice” sometimes at the expense of making the numerator messy, but for
comparison and other purposes that you will understand later – this choice is best.
Blitzer, Intermediate Algebra, 5e – Slide #19 Section 7.5