Square Root The square root of any real number is a number, rational or irrational, that when...
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Transcript of Square Root The square root of any real number is a number, rational or irrational, that when...
Square Root
The square root of any real number is a number, rational or irrational, that when multiplied by itself will result in a product
that is the original number
525 The RadicalThe Radical
Square Root
Radicand
Radical sign
• Every positive radicand has a positive and negative sq. root.
• The principal Sq. Root of a number is the positive sq. root.
• A rational number can have a rational or irrational sq. rt.
• An irrational number can only have an irrational root.
Model Problems
Find to the nearest tenth:
62 130
53824 4153
= 7.9
= 232
Find the principal Square Root:
4
164
9225 529= +15 = +23
2
1
8
3
180c = 13.4
= 11.4
= 64.4
Simplify:
x2 = |x|
4x16 = 2x8
(x2 2x 1) = x + 1= x
Index of 2
anradical sign
radicand
index
of a number is one of the twoequal factors whose product is that number
Every positive real number has two square roots
The principal square root of a positive number k is itspositive square root, .
k
has an index of 2
k2 k
kIf k < 0, is an imaginarynumber
Square Root
Index of 2
81 9
81 9
2 ?
Cube Root
Index = 3
Index of 3
anradical sign
radicand
index
of a number is one of the threeequal factors whose product is that number
( k3 )( k3 )( k3 ) k
k3has an index of 3
273 3 273 3
principal cube roots
nth Root
The nth root of a number (where n is any counting number) is one of n equal factors
whose product is that number.
kn
k is the radicandn is the index
is the principal nth root of k
325 2 25 = 32
325 2 (-2)5 = -32
6254 5 54 = 625
6254 not real
Index of n
anradical sign
radicand
index
of a number is one of nequal factors whose product is that number
325 2 325 2
principal odd roots
6 664 2 64 not real principal even roots
has an index where n is any counting number
kn
nth Root
Index of n
Radical Rules!
T
22522550
True or False:
41664 6464
T
25100 1010
25
425425
88
T
simplified
Radical Rule #1
In general, for non-negative numbers a, b and n
Example:
494936 623
x2 x3
baba
x3 x2
x5 x3
x5 x3 = x4
x5
x8
Hint: will the index divide evenlyinto the exponent of radicand term?
n n na b a b
3 37 2x x = x33 9x3 7 2x x
4 2x x x x
Radical Rule #2True or False:
25
b4 5
b2
5
b2 25
b4T TIf and
25
b4 25
b4
Transitive Propertyof Equality
If a = b, and b = c, then a = c
a
b
a
b
In general, for non-negative numbers a, b, and nExample:
144
81
144
81
12
911
3
then
n
nn
a a
b b 4
625
256
4
4
625
256
5
4
Perfect Squares – Index 2
12
12 144
11
11 121
100
10
10
9
9 81
8
8 64
7
7 49
6
6 36
5
5 25
4
4 16
3
3 9
42
21
1 1
Perfect Square Factors
Find as many combinations of 2 factors whose product is 75
751
155
Find as many combinations of 2 factors whose product is 128
323 Factors that are Perfect Squares
642
168
253
Simplifying Radicals
baba
80 answer must be in radical form.Simplify:
80Find as many combinations of 2
factors whose product is 80
1 80
4 20
2 40
5 16
8 10
5 16
16 4
perfectsquare
4 5comes outfrom underthe radical
•To simplify a radical find, if possible, 2 factors of the radicand, one of which is the largest perfect square of the radicand.
•The square root of the perfect square becomes a factor of the coefficient of the radical.
Perfect Cubes
13 = 1
23 = 8
33 = 27
43 = 64
53 = 125
63 = 216
73 = 343
(x4)3 = x12
(-2y2)3 = -8y6
3 48 3 38 6
3 48 32 6
Simplifying Radicals
3 48 answer must be in radical form.Simplify:
3 48 3 8 6
2) Express the radical as the product of the roots of the factors
3) Simplify the radical containing the largest perfect power (cube)
1) Factor the radicand so that the perfect power (cube) is a factor
Simplifying Radicals
Simplify: 33
4
33
4
36
8
2) Express the radical as the quotient of two roots 3
3
6
8
3 6
2
1) Change the radicand to an equivalent fraction whose denominator is a perfect power.
33 2
4 2
3) Simplify the radical in the denominator
33
4
Model ProblemsSimplify:
20KEY: Find 2 factors - one
of which is the largest perfect square possible45
45 5225
124 324344344 38
8021 5162
151621
5421 52
96
12
16 6
4 3
96
12
4 6
2 3
2 6
32
6
3
2 2
Model ProblemsSimplify:
4 345a b
3 616x
2 2 29 5( )a b b
23 5a b b
2 33 8 2( )x
2 32 2x
2 2 29 5( )a b b
2 333 38 2 ( )x