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Sometimes you need to rationalize the denominator of a fraction when the
denominator is a binomial radical expression. Multiply the numerator and
denominator of the fraction by the conjugate of the denominator.
Asi$ment 0uide
\f
Rationalizing Binomial Radical Denominators
oul".tive
tt:#:
46,48
Q
Extension 5'l-53
'
i+#
=#
1
_ (: + r6)(t + rA)
- (r - v5)(r + rA)
v7objecrive
fl
@ Core 13-26,3345,
47, 49-50
Q
Extension 54-55
7/ttanUardized
dffi.
Rationalize the denominato,
@@Core 1-12,27-32,
_8
--t -4
Mixed Review 63-76
#
Yisual Learners
€l
Erercise 47 Suggest that students
sketch a diagram for this exercise.
4\/5
-4
cl-Fs,'
Examples 1 and 2
(page 374)
3
(page 375)
Add or subtract if possible.
t
Example 4
(page 37s)
cannot combi
axG
4.3\/tr - 5\,G -2f,
s\,6 + t/6
)-
- zl/1 +fG
s. M\/i + 3\5
2.6Vt
3.4\/3 + 4v
6.71G
cannot combine
Simplify.
7.
Multiply.
z:l/;'
5w
- 3\,1813\6 9. r,4s + ln t
n.3l-ll - 2f,q
D.{i + {qs
zfli + zffi
sf? - af6.
rs.(3+ 16)(r + \,5) 8 +4fa
- 6\0) 63 - 3BV2
(V13
17.
+ 612 4s + 12V1s
rs.
-
3
6\fE + 3ym 3g\,4. s. r4\/n
n.VA + VG sf6.
(.1
-
4vz)(s
M.(z+ t/1)(r+zxe) zs+i
I
+ 2\/15
16.
(\,€ + r/s)2
18.
(2Vs + 3\/2)2 s8 + 12\ x
! (v; r5)1f+$)
"#*
- i.?:-*.,.*...'"."",
"**
Example
5
(page 375)
-'l:Y"
Multiply each pair of conjugates.
19. (s
- vlT)(s + \/n) A
21. (2\,G
Example 6
(page 375)
376
Chapter
7
+ 8)(2\/6
-
8) -40
-
z\/i)(4 + z\,3) 4
22. (\,5 + V5)(V3 - \5) -2
20. (4
Rationalize each denominator. Simplify the answer.
n'
r+a-
24.# _.
2
26. See margin.
26.?:#
,<5+\/J
Radical Functions and Rational Exponents
376-378
2s. -2 + 2VE
pages
24. 12\/+!_B
i375
39 + 1o\,4E
For more practice, see Extra Pr
eractice by Example
Example
--
- r,6.
Simplify.
CheckUrrdarsfandihQ @Rationalize the denominator
EXERCISES
i,..-1t
or 1
V5
Error Prevention
Erercises 7-12 Remind students
to first simplify the radical
expressions they want to add or
qrbtract. Otherwise, they will not
always find all the like terms.
r
is the conjusate
Multiply.
3+3\B+"rB+
Test Prep 5G-G2
+ \/5
Exercises
25. 13 +
26,
7fi
1t+#
)
Apply Your Skills
Simplify. Rationalize all denominators. Assume that all the variables are positive.
27:\m +
See margin.
2s.
s\Ei
+ vts 13\,
+ 4\Aw 4s\ t
v32
30. v?5
-23s
x
-
t/z
44-4 -_2,!z
+,s{F
Vq
2\3
Quiz?'3
Lesson
Simplify.
1. 6V10
7\A)
-
4vlO 2\ m
slG + 59x 13\3i
3. 2\,8 + \,48 10\ 3
- r48X: + r,4E)
38. (2\/t - zt/1)(+^,/y - s\,A)
40. 4.t Y6,, 2\ 3 - \ t
vl
vJ
+,y z*:f,G
42.3
I
ag + +2.Y'g
Ax
(2\,5 + 3\/2)(sl/5
fu
2.
36. (2
(\,G + .,t3)(xC + 2\,3)
$.5 +,{
(-BC-1+\/5
D-BD_
2
4\48 + 2\,52
34.
At.mfrifr
found in
rlh patterns of many
avg
- \,1N^,5 + 2\,5)
3s. (1 + ln)6 + \A)
(v5
2n 4 \/E
"''Z-ZyE
is
-
- sr6
32.3f/G-4f/s4+flns
37.
len Ratio
\/15 + 2\/48
31.4xEG + 3\Ef
33.
uEft-Sen-n*ti-oJ
28.
4.i/qn -
ftri tE
5. Multiply (Z + Zr,€)(S - '' :
29 + 3\,5
6. Rationalize the denomina::'
'
2 _ \312
16-lU'7
inL2\].
Yo
10-v7
1 +z\/5.Find
the reciprocal of the Golden Ratio and
compare it with the number that is I less than the Golden Ratio. See margin.
45. The Golden Ratio is
46. Critical Thinking Describe the possible values of a such that x42 +
be simplified to a single term. a must be twice a perfect square.
ffi
K
.\
t4
qZ. Geometry A rectangular walk is
m wide and
6t/i
tG
can
m long. What is the
perimeter ol the walk? 14V7 m
48.
Writing
Discuss the advantages and disadvantages of first simplifying
t/i2 + t/i + Vf S in order to estimate its decimal value. See margin.
3. See margin.
.v21 ttts.How long will it take
49. Open-Ended Find two pairs of conjugates with a product of
S
fl
chutt"ns"
SO.
Physics An object is moving at a spe_ed of (3 +
the object to travel 20 ft? 60 - -2oy 2 s, or about 4.53 s
_1
52.
2
53. Forwhatvaluesof aand
bdoest/a +
V)VJV)IVJ
-l-
\,6:
As-.sessm e n
t
Have students write two addi:t cproblems and two subtractioproblems that involve radica s
and that have an answer o'3'. J
Then have them write a mu t plication problem and a di', s c*
problem, each involving t'r"t
binomial radical express'o-: --- =:
have an answer of 2 - '.5 -=.:
students share their proc =-:
with another student tc i'e:{i
their work.
Add or subtract.
5t.
""l-v5..l = . __L_
1+V5
*llglqativ-g
+__- 4\ 3
\'G + b? (a = 0andb > 0)or
(p =-0 and a > 0).
54. Error Analysis A student used the steps shown below to iimplify an expresSion.
Find the student's error and explain why the step is incorrect.
|
rr
0
- \/2f
-
ln the second step the
exponent was incorrectly
distributed: (a - hl* * ax
2
'ttt-
......................................::::--ll- vLl
: t_z _ (r/2) ,
1
_1_
_
45. The reciproca
- ff.
-,+$,
rz 6,ry)
less than
-L-L2
-1
_1
-2
and n can be any
csitive integers.
must be even or
must be odd.
r
55. In the expression
{in
,
m
and
n
are positive integers and x is a real number.
The expression can be simplifled.
a. If -r > 0, what are the possible values for m and n?
b. If x < 0, what are the possible values for m and n?
c. If x < 0 and an absolute value symbol is needed in the simplifled expression,
what are the possible values of lz and n? m must be even, and n can be
any positive integer.
Lesson
?-3
Binomial Radical Expressions
377
s
whrc.
1-':
s cre
:
,
48. Answers may vary.
Sample: Without
simplifying first, you must
estimate three seParate
square roots, and then
add the estimates. lf they
are first simplified, then
they can be combined as
13\ 2. Then only one
square root need be
estimated.
49. Answers may vary,
,i - q\,6
y\'6
33. -11 +\E
34. I + \,/iO
2\3i
s5. 17
+
g1.V?
- $ft
s7.x+svt+o
36. -36
38. 8y
- zzfzy + So
Sample:
NE+zlWE-zl
Pr6.+x,GXzxh-r[O
377
hsources
practice with
Multiple Choice
a
-
Which expiession does NOT simplify to one term? B
57.
What
56.
Practice, p.419
p.414
\E - ltr
- \/15
r,4a
3V8
B'
D' \'42
A. -4\/tr' +
c.\,8 +
of test item formats:
Daily Practice and
F.
ies Transparencies
58.
is an expression
\,65
for v26
G.
13\5
-
\-80 + t/ 1-25?
-l
H' 11\6
Which expression is NOT equal to 13? D
+ v5X4 - \€)
c. (e + V23Xo - \E)
A. (4
58 Point out to students
the products are Products of
They should be able
incorrect choices
actually finding all terms
59. How can you write
F.
f$
G.
-1
50. Which of the
following
understanding of
7-11o7-3=
Extended ResPonse
61. lsthe product
is
I
- 3\/G
r.4j
equivalent
(r - VeXr * V8)
62. What is the value of
- zt'tr)(s + 2..-t7t
D. (7 - x6)(t + t/6)
B. (5
with a rationalized denominator?
H. 4 + 3V3
to
8.6+9\5
A.51
Short ResPonse
-2
l' -3rE
(2 + 3!5X3 + 31/5)? D
D. 5l
c. 6 + 24\/E
a rational number? Explain'
t+A - f]6t
Show your work' See
int Quiz 1 (p. 378)
Teaching Resources
nt Quiz 1 (also in
lesson
Prentice Hall Assessment
7-2
that
simplify each expression. Rationalize all denominators' Assume
are
System)
positive.
\5i
$.{t .flts sf,r64. \,6i'^fTS
leaching All Students
67.1,5*.fl4x
and Math LiteracY 78
2x
versions available
Lesson 6-4
71. 2,
-1
X
1-4=-3,whichis
rational.
value -3 onlY OR
the
[1]
yes only OR minor
error in calculation
71.2x3-16:0
74.x4-14x2+49:o
i\/4
tfl
a
(F
lnstant self-check
qulz onllne ano
on CD-ROM
pl
\/8(\/r4 + 3\€) ry's +
z+
3W e*4 - gf,i
9.(\/5 + 2\/3)(\/5 - z','5) -z
Chapter
7
Radical Functions and Rational Exponents
[2] correct method, but
finds 1ei#y2 +
E+#
minor error, but
appropriate method
g
-10,5 t sif
:
0
72. x3 + 1000
75.25xa - 4ox2 + 16:
xzf s
-il
'0.i+
1 -1 +i1.3
5' -- tE-
73.125x3
o
76.8Lx4
- l:
- t:
+6,t1
t.{FF lblc2 z.fl}F xvz s.fFF -a
j.4{Bj, -
378
6\-2
n
when
Simplify each radical expression. Use absolute value symbols
s.
s+x,6.
s+x6.
-,/- 5+2.V2
- ^^/^
s-2\/2
15 + 6\/t
10 - 4\'t
=E-44-E=A:t
- 10 !N2-lE17
6s.ffi :/sn
are Positive.
2 .5-2vzs-zt/z
17
6u.*
+
4
Assume that all
Simplify each exptession. Rationalize all denominators'
s-2\4.-
fi - 4\/,
thx . YI4x'
u2\6. -
Solve each equation.
376'378 Exercises
14 (1 - S6ttr + f61 =
(1 -2X1 *21 =
tltt2-31'+r;;i6
68.
es.ffi
al
zs + x,6.
and sets
[1] answer onlYn no work
shown
6.2\EF
.zfnFf
qtfiw
^ 4\'2ii
-6
t.
- r,
9V Jx'Y
s ---87zor,/z - zs
ru.07fi,

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