Einstein's famous formula E : mc2 relates energy -E, mass m, and.the speed
of
c. Express c in terms of E and m and,rationarize the denominator.
light
-1
L:MC'
E
^z_
L-
c:
Assignment 0uide
l/obiective
m
E; lEu :
IE
tlA
\,1
nck Uyrdgfsfafrding @ rne formula a
t/^2
_2
@ @ Core 1-22,374s,
56-58, 59
Ye*
m
: 4 relatesthe accelerati on a of amoving object to the distance d it
moves in the time'l. Solve the formula for / and rationalize the denominator.
S) Extension G1-63,68
f,Tohiective
Y
f= lau
a
$QCore
23-36,46-54,
55, 60
IERCtSES
For more practice, see Extra practice.
$) Extension 64-67
Standardited Test prep 69-75
Practice by Example
Example
1
1.
(page 358)
s.
4
tli + 3.1G . flA -s 4.{G. 46
6. f/=. f/-zssl.l/s. V4 -6 s.V42 f/4
vE. tlzz rc z.f/ 4.
v:5. \6
not possible
Example
2
(page 369)
Example 3
(pase 369)
s.{zo,l
tt. {s+l:
(page 370)
Example
5
(page 370)
12.
L[.\,moott $.v _2su(,j
.29ifi 3ov2f,E
22.
Divide and simptify. Assume rhat alt variables
-lDx2y2
use
l6as
16.{6W
Connection to Physics
Exercise 35 You may wish to
point out that the distarrce r b the
distance between the centers of
the objects.
.2{rsx,
ar-"i:tt*F
zr.ffro ,l.ffiV ,s.W ,t.ffP
Rationalizethedenominatorof each"*.".rroo??ri{rf"rn^r"uo"rfliLff
#
are positive. 30-34, See margin.
2
#
V5
3r. ^4/;
(page 371)
r. \EoF
9-16 Remind students
absolute value signs since the
variables are positive.
rcaso\,6 -sx2v@
zvfFF
.
^..gt@
Multiply
and simplify. Assume that all variables are positi've.
rz.'V 6 .lG zfli
ffi. V8)f . f 4oF sf{w
rs.fiF .f 4r.j zff{av
20.4\Oi .s{6x} aoxyfs
\'-u
,,
-'. \,8
\rO
Exan'lple 6
n.f/8Lf
Error Prevention
Exercises
that they do not need to
Simplify. Assume that all variables are positive. 9-12. See margin.
zu.3grt
Example 4
Mixed Reuiew 76-95
simplify.
Mulfiply, if possible. Then
2s.# !;Y
ro
^3/
-\14
r, 1s\40'5
"-'
3\42x
--
v
35. Physics The formula
F
:
t/
G^t*'
r'
,,
v2
9E
_T
^-
30.v*
^ /34.9
'xv'
5*y3
\/2*2y3
relates the gravitational force Fbetween
an object of mass m1 and an object of mass m2separated by distance r.
G is a constant known as the constant of gravitation. solve the formula for
Rationalize rhe denomina,or.
r.
,, = IS+!r,
r
Lesson
f371-373 Exercises
12. 2aV4a2
\r'Sx
7-2
Multiplying and Dividing Radical Expressions 371
32, sx2\,G
t_
fgx2
30.
f,@
--d-
33.
\'t
31.
*6
-5-
34. xvqo
fev
5y
-T-
371
fl
npp[vour skills
36. a. Simplify
V#by
multiplying the numerator and denominator tt).
b. Simplify the expression in (a) by multiplying by
- t/z
f.
L
Lesson
'6
)ti'ff
c. Explain how you would simplif y
8ulz7'2
rA
instead of
r5'
a-c' See margin'
Assume that all variables are
simplify each expression. Rationalize all denominators. Assume that all var
positive.
are positive.
1. Multiply. Simplify if possible.
3i. \,5 .
a. r,6 .t/+s ls
b. fl4 .flzooo zo
+o. zox2f\,G
40.
7ft
42. 15 a slx
$. \/5(\,5 + r,45)
41. 10 +
2. 5implify.
Ve"s 2x2xEi
b. V -2laFim -3xrF
".
It
3. Multiply and simPlifY.
44.
.\,8?F
".{TeF
6x2vt/ry
tVio6t
+s. s +
ou.
s\,6
fln,4,
v
Y
ffi
v3.'
{F
b.w
V4z
nee!.WsaC$!9.n.nection
A satellite being launched
from the cargo bay of the
S
Have students work in Pairs.
Each student should make uP
cn"lt"ng"
-2Y
I2x"
i
-
1
+s
Se.
Geometry A rectangular shelf is t/440 cmby
rE
cm' Find its area'
physics A freely falting object hit the ground i" \fiS,f seconds.It fell
€F St.
Use the formula n : {et2 io flnd /z in terms of a' 288 a5 tt
%
\
\,f : f/?
Seemargin.
ss. Writing Does
67.a=-2c,b=-6d
4+r
Sample: First simPlifY
372
7fi,to
rationalize the
denominator, mu|tiPlY
\
the fraction OY ?
\/''
This yields
zz*:62=2!fi14
and
.z
f
'
which do
1
ar.'[E@
zrv
64.ft;1;-
w
67. Critical lhinking Wt en
o.{{@
uvz
:
es. Vi/sooo
66.
Vtf
is simplifled, the result is
va
zr
{*
aTa
V-= - rn
.,L,
where
r
are positive integers. Express a in terms of c, and b in terms of d'
ti-1
68. Criticallhinking In Example 3 you saw tnlL-fl S+?sF' VS"f
z*yzfl tofy,if you assume that all the variables are positive' Now as:
the variables represent any real numbers' What changes must be made
answer? Explain. See margin.
,.
v' \/6+3
15
c, Answers may vary.
the denominator.
Since tG =
i.i
Vi'#'
simplify each expression. Rationalize all denominators. Assume that all
other's work.
pages 371-373 Exercises
for all, some, or no values of x? Explain'
arp positive.
a problem similar to those in
each of Examples 1-5. Then
have students work each other's
problems and check each
372
f'5? ',fl]
*.-tv* lq.,"
t4I*
5l----:4s.
prefertouseforflndingadecimalapproximationwithacalculator?Ju
yourreasoning. Check students' work'
Alternative Assesslnent
T\22
{2.f
so.E
Y7x'y
60. Open-Endled Of the equivalent expressions
space shuttle
',
.f,q
\/:t f=g: l4trq :
^ tta f21,
249 =
.t6]
+t.f11r uYP
44.f,[2x
57. Error Analysis Explain the error in this simplification of radical expres
\/Tn = 4 56-157' See marsin'
'
expression.
\
t)
in miles from the satellite to the center of Earth. How much greater is t
velocity of a satellite orbiting at an altitude of 100 mi than one orbitilg
greatet
altitude of 200 mi? (The radius of Earth is 3950 mi') 212 mi/h
5. Rationalize the
denominator of each
su".
at. vA(/SA +
s4.\/1_! \i
s3.3 sl65
s2.-z(fE*fls+)
-4yi - f,E
hour, of a satellite orbitir
€F sS. Saieitiies the circutar velocity u i1-1:M.
\012,
\r
where r is the distance
Earth is given by the formula v : ^lt'24
lfiw sf,t
"'fni,7
q,
-r,6gJ
n +fG
1 + \A
z'vVsG
u
3s.l/ 4 'fl
+o.ffirv
x*6.
4. Divide and simplify.
- r,lnF er\6
"'-Jr;
:f:+
Y2x'y' 2y'
+5. sxzf/x
.
rcxE
s\,5# .rft6,
ttlrc
sx6f\zy
u' Vxsf ' :r :
42.3(s + \/n)
€hapter
7
Radical Functions and Rational Exponents
59' For some values' it is
easY to see that the
equation is true if x = 0
56, 2O{22 cms
57. A product of two square
roots can be simplified in
this way only if the square
roPlq are ref+umbers'
V-2 and V -8 are not.
x = 1' But when
x < 0. f,F i" not a real
or
numbe[ atthough
97
i".
68. No changes need
made; since theY i
odd roots, there is
need for absolute
symbols.
Multiple Choice
69. Which expression does NOT simplify
A.
c.
"'W
.9C
2xy
71. What is the simplified form of
o+r
,Veb Code: aga-0702
-
L'-
3\/tr
variety of test item formats:
with a rationalized denominator?
ffi
*"
.
H
,''W
fffz
,
wrv
zxy
Strateg ies Tra nspa rencies
Exercise 59 Tell students to
answer choices, paying close
B.#
- \,8
FCAT Practice, p. 419
Strategies, p. 414
FCAT Daily Practice and
carefully examine all of the
a
attention to the
74. 12! fi
D.14_ffi
--
R"rour.",
For additional practice with a
a-
B.
E_
-v25 .v -32
''
PHSchool.com
Fl
C
\/25.V-8
D. ff-tx .f,ta
-f/1ooo
70. How can you write
brmat quiz at
to - 1O?
real number if x
ffi, uy what number would you multiply
the numerator and denominator of the fraction? G
F.2
G.3
H.6
A.f
2oxz
B.
Vsix
5hort Response
74. For what values of x is
nded Response
75. Rationalize the denominator of
c.
t& . \/ -,
form?
[1] answer only OR error
D,+
a real number? Explain.
i,E.
makes Vx.\ -x a
real number is x = 0.
D
Vro
describing value(s) of
variables
74-75. See
margin.
E"ptain your steps.
0,
So the only value that
t.9
73. Which of the following expressions is in simplest
i" a real number
ifx>0and\ -xisa
<
72. To rationalize the denominator of
/
signs.
75. l4l You should muttipty
ot,E
o""rur"
4x'
\ iA<
"F
Y
14,,
"E
zx Aqr2
f€ f,EF
\'l
Lesson
7-1
Simplify each radical expression. Use absolute value symbols as needed.
76.$zlF
11laasl
77.
6-3
<-
79.V 32yzr 2y5
so.fotiF
u..w\!S
o.5lx3l
which has a
denominator without
a radical.
,zrs
$.\4-ooo+"zo
o.o8x2o
[3] appropriate methods,
but wiih one minor
Divide. Terr whether each divisor is a factor of the dividend.
84. (y3
8!:
-
64)
+ (y +
o,
a)
85. (x3
- !+4)-:(2a+I)
^6!t_*
84-87.
See margin.
Lesson 5-7
{zx \o 4x2
V1u? _ ? tuz
Wr, - ,-
_g.2a432
7s.V:64p- -4"2t
82.*I6#6y% 2l*sly2a
Lesson
*\/6r4W
+
87.(2xa
-
27)
error
+ (x +
3x3
3)
[2] major error. but
subsequent steps
consistent with that
error
-4x+10;)_(x_2)
Complete the square.
88. ,r2
+ 10x +
ri$
25
90.x2+Ilx+:*ff
e2.x2- i** *
94.x2 - J, + *t
*e
tesson
[1] correct final
- 10x + iii 25
91.xz-11x+,* +
g9.
93.
95.
7-2
x2
expression, but no
work shown
x2 + 0.3-r + iit* O.O225
x2 +
J; + !|
84. y2
,*g
Multiplying and Dividing Radical Expressions
-
4y + 16, R -12e,
-
3x + 9, a factor
not a factor
85. x2
973
86, 3a2 - 8a
factor
87,
-
2, R6,nota
2f + x2 + 2x, R 10, nota
factor
373