Download q48=q( 2* 2 * 2 * 2 * 3)

Radical Expressions and Equations
The Media nies: Simplifying Radicals
A radical expression is simplified when
1. The radicand has no pcrfectsquare factors other than I
2. The radicand has no fractions
3. No denominator contains a radical
jif in the violations
Determine which condition is violated for the following radicals.
a, “14$
2
b. ‘1’7n
C.
f. J12
g. ‘132
Ii
iIi
‘36
2
J6
d.
5
,J96
/l2
6
L27
1 ‘‘F’’’
PrfçS.uareFactoi
According to condition 1, the radicand should not have any erfectsiuare
1. If it does. then
• Write the radicand as a product of prime factors.
• Identify an perfect squares to be removed.
• Remove perfect squares, leaving, non-perfect squares a radicands.
“
V3
R)f S
other than
L4
Examples:
82
=q(
4
q
*
2
*
2
*
2 * 3)
=[(2*2)*(2 *2)*3]
j[(4)*(4)*3]
j(42*3)
=4I3
2 *2 * 5 *
*
2) * 2 * 5 *
= i[(2
=J(2L*2 *52)
2 * 5 2
=lOd2
J2OO
=
i(2
*
write as product of prime fctor
group together pairs of duplicated
multiply pairs
write as a number squared
pull out perfect-square
write as product of prime factors
group together pairs of duplicated #s
write as numbers squared
pull out perfectsquares
Simplify
Practice: Removing Perfect Square Roots
a.
b. ‘5()
The Mechanics: Simplifying Radicals
c. I96
1
d 243
3
OO
Z cX
c)
Radical Expressions and Equations
Simplifying Radicals: Cib
-
Removing Variable Factors
• As with numbers, variables are treated the same way
• Shortcut:
o if variable is even power. then pull variable out and raise it to ½ the power
o if variable is odd power, then pull variable out & raise it to (n I )/2 power &
leave variable to 1 power in radicand
Examples
5
‘a
a * a * a * a)
*
=
*
*
*
a) (a a)
) * a]
2
(a
*
=
JJ(a) a]
*
2
aa
3
q27a
*
a]
write as product of prime factors
group together pairs of duplicated #s
multiply pairs
write as a number squared
pull out perfectsquare
3 * 3 * a * a * a)
3) * 3 * (a * a) * a]
..J[2) *
3 * (2) * aJ
3a3a
(3
*
write
*
=
-
as product of prime factors
group together prs of duplicated #s
write as a number/variable squared
pull out perfectsquares
Shortcuts:
=
*a
4
i
1
\
=
a= a2
as even power * odd power
pull of even & leave variable to 1 power in
write
a
a
pull
of even power out
Practice: Removing Variable Factors
3
a. J16a
b. 45a
c. 28a’
Simplifying RathcalsC1cujjin Radicals
When 2 radical expressions are multiplied together, either
As I radicand. write each radicand as a product of primes
Or
• Use the multiplication property of square roots
In either case, follow previously state guidelines to simplify under the radical.
The Mechanics: Simplifying Radicals
Radical Expressions and Equations
Exampies
Method 1:
‘418
*
112 J(2*2*2*2*2*3)
wilte as produut of prime factors
[(2 *2)*(2 *2)*2 *3]
i(22) * (22)
1[(21)2* 2 *
4’41 6
3’2b
4’JlOb
group togeehci pis of dupicited “s
multiply pal r
vritc as a number s4uared
t’oar
rul ut p
*2*3]
31
J
L
Mult. whoie
miit
ois
Write as prod of primesperfrce squarc
Simplify vi2Lbi
Simplify
2
l2’41(2Ob
= i2(22 *5 * b
)
2
= 12 2h415
= 24b’415
Method 2:
18*112 =\96
Use multiplication property of square ioo
i(l6*6) Simplify under radical
I16
Use multiplication property of square roots
Simplify l6
Practice: Multiplying Radicals
a.ii3*J52
b.4l2*432
c 5’v3c
*
\bc
d. 25x
thinRadI
According to condition 2, the radicand should not have an frauk ns if i
the fractions from the tadicand by
Usmg drision property of squaw roots
Or
• Dividing
Examples (Using Division Property):
r;
4
2
rOOtS
‘4
i0
Simplify
—
(4
Practice: Removing Fractions within Radicals using Div Prop
a.
/11
49
13
.
\
i44
9
—
The Mechanics: Simplifying Radicals
c.
3
/13
64
j-—
a
‘U
3
6’1Ox
hen remove
[fc;
Use division property of square
*
Radical Expressions and Equations
Examples (Dividing):
= ‘J8
Divide
ç4*2)
=2I2
Write as product of primes/perfict squares
1.
2.
l2a
27a
=
Simplify
[4u2
\
Divide numerator & denominator by ia.
9
Use Division Prop of Square Roots
9
,t
=
Simplify
Practice: Removing Fractions within Radicals by Dividing
a.
b.
c. J--——
3x
75
d.
‘hi48x
ar
According to condition 3, the denominator of a fraction should have no ra eai if it does,
then rationalize the denominator by
• Multiplying both numerator& denominator by
such that thr erominator
nurnüer
Jmmator
would become a pertect square. (Hint: usually ===
)
qdeno mm ator
•
Use multiplication property of square
roots
• Simplifr
Examples:
7
,,
%J5
q5
*
=
=
25
—
5
Multiply by
—
to make denominator perfect square
Use Multiplication Property of Square Roots
Simplify
Practice: Rationalizing the Denominator
a.
3
--
J5
U.7
The Mechanics: Simplifying Radicals
$
f7m
1
d