radicals are in simplest form when: no factor of the radicand is a perfect square other than 1.no...
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Radicals are in simplest form when:
•No factor of the radicand is a No factor of the radicand is a perfect square other than 1.perfect square other than 1.
•The radicand contains no The radicand contains no fractionsfractions
•No radical appears in the No radical appears in the denominator of a fractiondenominator of a fraction
Perfect Squares
1
4
916
253649
64
81
100121
144169196
225
256
324
400
625
289
Multiplication property of square roots:
Division property of square Division property of square roots:roots:
baab
b
a
b
a
4
16
25
100
144
= 2
= 4
= 5
= 10
= 12
To Simplify Radicals you must make sure that you do not leave any
perfect square factors under the radical sign.
*Think of the factors of the radicand that are perfect squares.
8
Perfect Square Factor * Other Factor
= 2*4 = 22
LEA
VE
IN
RA
DIC
AL
FO
RM
20
32
75
40
= =
=
=
5*4
2*16
3*25
10*4
= =
=
=
52
24
35
102
Perfect Square Factor * Other FactorLE
AV
E I
N R
AD
ICA
L F
OR
M
48
80
50
125
450
=
= =
=
=
3*16
5*16
2*25
5*25
2*225
=
=
=
=
=
34
54
25
55
215
Perfect Square Factor * Other Factor
LE
AV
E I
N R
AD
ICA
L F
OR
M
If you cannot think of any factors that are perfect squares – prime factor the radicand to see if you have any repeated factors
EX: 20 20
2 10
2 5
52
You can simplify radicals that have
variables TOO!
2x xx* == x4x = xxxx *** =
2x
46 yx =
Radicals are in simplest form when:
•No factor of the radicand is a No factor of the radicand is a perfect square other than 1.perfect square other than 1.
•The radicand contains no The radicand contains no fractionsfractions
•No radical appears in the No radical appears in the denominator of a fractiondenominator of a fraction
Multiply Square Roots
REMEMBER THE
PRODUCT PROPERTY OF SQUARE
ROOTS:
baab
abba
OR
Multiply Square Roots
To multiply square roots – you multiply the radicands together
then simplify
2EX: 8* = 8*2 = 16
Simplify 16 = 4
Try These :
2 2
9
*
* 4
2 18*
Let’s try some more:
2
10
65
5
10
3 12
Multiply Square Roots
•Multiply the coefficients
•Multiply the radicands
•Simplify the radical.
Multiply & Simplify Practice
)32( )104(
)25( )38(
)54( )3(
)42( )53(
)103( 59
Homework Practice
)33( 52
)34( )105(
)25( )38(
)53( )2(
)46( )57(
1.
2.
3.
4.
5.
Radicals are in simplest form when:
•No factor of the radicand is a No factor of the radicand is a perfect square other than 1.perfect square other than 1.
•The radicand contains no The radicand contains no fractionsfractions
•No radical appears in the No radical appears in the denominator of a fractiondenominator of a fraction
Division property of square Division property of square roots:roots:
b
a
b
a
To simplify a radicand that contains a fraction –
first put a separate radical in the numerator and denominator
25 25
36 36
Then simplify25 5
636
Try These :
16
225
100
64
16
36
49
81
Simplify:
If we have a radical left in the denominator then we
must rationalize the denominator:
20
36
20
36= =
Since we cannot leave a radical in the denominator we must multiply both the numerator and the denominator by this radical to rationalize
52
6
52
6* 5
5=
5*2
56
10
56=
Simplify: Hint – you will need to rationalize the denominator
A. B. 4 C. D. 16
5
80
Simplify Radicals
1) 1)
2)2)
3)3)
4)4)
2
6
5
3
2
43
8
6
5)
6)
7)
8)
3
92
32
10
3
185
2
8
Simplify some more:
Review writing in simplest radical
form:
1)2)3)4)
64sr
236x
23
29
100
14264 pn
5)
6)
7)
8)
9)
36
7
4
16
25
4
Review writing in simplest radical
form:
Which of the following is not a condition
of a radical expression in simplest form?
A. No radicals appear in the numerator of a fraction.
B. No radicands have perfect square factors other than 1.
C. No radicals appear in the denominator of a fraction.
D. No radicands contain fractions.
Adding and subtracting radical expressions
3 3 and 5 3 are like radicals
•You can You can only add or subtract radicals only add or subtract radicals together if they are like radicals – the together if they are like radicals – the radicands radicands MUSTMUST be the same be the same
3 3 and 3 5 are not like radicals