§ 7.5 multiplying with more than one term and rationalizing denominators
TRANSCRIPT
§ 7.5
Multiplying With More Than One Term and Rationalizing Denominators
Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.5
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 #3 Section 7.5
Multiplying Radicals
EXAMPLEEXAMPLE
Multiply: . 111023(b)4763(a) 333
SOLUTIONSOLUTION
Use the distributive property.
333 4763(a) 3333 47363
Multiply the radicals.
33 12718
111023(b)
112102113103 Use FOIL.
22203330 Multiply the radicals.
Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.5
Multiplying Radicals
22543330 Factor the third radicand using the greatest perfect square factor.
CONTINUECONTINUEDD
22543330 Factor the third radicand into two radicals.
22523330 Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.5
Multiplying Radicals
EXAMPLEEXAMPLE
Multiply: .2(b)73(a)2
33 yxxx
SOLUTIONSOLUTION
Group like terms.
Multiply the radicals.
Use FOIL.
73(a) 33 xx
7337 3333 xxxx
2137 333 2 xxx
2137 333 2 xxx
Combine radicals.21433 2 xx
Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.5
Multiplying Radicals
Multiply the radicals.
Use the special product for
22(b) yx
CONTINUECONTINUEDD
22222 yyxx .2BA
yxyx 222
Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.5
Rationalizing Denominators
EXAMPLEEXAMPLE
Rationalize each denominator: .16
10(b)
5(a)
5 23
2xy
SOLUTIONSOLUTION
(a) Using the quotient rule, we can express . We
have cube roots, so we want the denominator’s radicand to be a perfect cube. Right now, the denominator’s radicand is . We know that If we multiply the numerator and the
denominator of , the denominator becomes
3 2
3
32
5 as
5
yy
2y.3 3 yy
3
3 2
3
by 5
yy
.3 333 2 yyyy
Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.5
Rationalizing Denominators
The denominator no longer contains a radical. Therefore, we
multiply by 1, choosing .1for 3
3
y
y
3 2
3
32
55
yy
CONTINUECONTINUEDD
3
3
3 2
3 5
y
y
y
3 3
3 5
y
y
Use the quotient rule and rewrite as the quotient of radicals.
Multiply the numerator and denominator by to remove the radical in the denominator.
3 y
Multiply numerators and denominators.
Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.5
Rationalizing Denominators
CONTINUECONTINUEDD
y
y3 5 Simplify.
(b) The denominator, is a fifth root. So we want the denominator’s radicand to be a perfect fifth power. Right now, the denominator’s radicand is We know that
If we multiply the numerator and the denominator
of , the denominator becomes
5 216x
.2or 16 242 xx
.225 55 xx
5 3
5 3
5 2 2
2by
16
10
x
x
x
.22222 16 5 555 35 245 35 2 xxxxxx
Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.5
Rationalizing Denominators
CONTINUECONTINUEDD The denominator’s radicand is a perfect 5th power. The
denominator no longer contains a radical. Therefore, we
multiply by 1, choosing .1for 2
25 3
5 3
x
x
5 245 2 2
10
16
10
xx Write the denominator’s radicand
as an exponential expression.
5 3
5 3
5 24 2
2
2
10
x
x
x Multiply the numerator and the
denominator by .25 3x
5 55
5 3
2
210
x
x Multiply the numerators and
denominators.
Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.5
Rationalizing Denominators
CONTINUECONTINUEDD
x
x
2
2105 3
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.5
Rationalizing Denominators
EXAMPLEEXAMPLE
Rationalize each denominator: .3
3(b)
37
12(a)
xy
yx
SOLUTIONSOLUTION
(a) The conjugate of the denominator is If we multiply the numerator and the denominator by the simplified denominator will not contain a radical. Therefore, we
multiply by 1, choosing
.37
1.for 37
37
,37
37
37
37
12
37
12
Multiply by 1.
Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.5
Rationalizing Denominators
CONTINUECONTINUEDD
22
37
3712
Evaluate the exponents.
22 BABABA
37
3712
Subtract. 4
3712
Divide the numerator and denominator by 4.
4
3712
3
1
37
37
37
12
37
12
Multiply by 1.
Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.5
Rationalizing Denominators
CONTINUECONTINUEDD
Simplify. 3373or 373
(b) The conjugate of the denominator is If we multiply the numerator and the denominator by the simplified denominator will not contain a radical. Therefore, we
multiply by 1, choosing
xy
xy
xy
yx
xy
yx
3
3
3
3
3
3
.3 xy ,3 xy
.1for 3
3
xy
xy
Multiply by 1.
Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.5
Rationalizing Denominators
CONTINUECONTINUEDD
xy
xy
xy
yx
xy
yx
3
3
3
3
3
3
Multiply by 1.
22
22
3
323
xy
yyxx
Rearrange terms in the second numerator.xy
yx
xy
yx
3
3
3
3
222 2 BABABA 22 BABABA
xy
yxyx
9
69
Simplify.
Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.5
Rationalizing Numerators
EXAMPLEEXAMPLE
Rationalize the numerator:
.7
7 xx
SOLUTIONSOLUTION
The conjugate of the numerator is If we multiply the numerator and the denominator by the simplified numerator will not contain a radical. Therefore, we
multiply by 1, choosing
.7 xx
1.for 7
7
xx
xx
Multiply by 1.
,7 xx
xx
xxxxxx
7
7
7
7
7
7
Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.5
Rationalizing Numerators
Leave the denominator in factored form.
xx
xx
77
722
CONTINUECONTINUEDD 22 BABABA
xx
xx
77
7Evaluate the exponents.
xx
77
7Simplify the numerator.
xx
7
1 Simplify by dividing the numerator and denominator by 7.
Blitzer, Intermediate Algebra, 5e – Slide #18 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
22))(( BABABA
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 #19 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 bythe 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 forcomparison and other purposes that you will understand later – this choice is best.