Transcript Aim: How do I tame radicals? - Troup 6

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
The Radical
Radical
sign
25  5
Square
Root
Radicand
• 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:
c  180 = 13.4
62 =  7.9
130 =  11.4
53824 =  232
4153 =  64.4
Find the principal Square Root:
225 = +15
1
4
529 = +23
9
64
1

2
3

8
Simplify:
2
x
x ==|x|
4x
16
=
2x8
2
(x  2x  1) = x + 1
Index of 2
index
radical sign
n
Square Root
Index of 2
 81  9
81  9
2  ?
a
radicand
of a number is one of the two
equal factors whose product is
that number
has an index of 2 2 k  k
Every positive real number
has two square roots
The principal square root of
a positive number k is its
positive square root, k .
If k < 0, k is an imaginary
number
Index of 3
index
radical sign
n
Cube Root
Index = 3
a
radicand
of a number is one of the three
equal factors whose product is
that number
( 3 k )(3 k )(3 k )  k
has an index of 3
3
k
principal cube roots
3
27  3
3
27  3
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.
k is the radicand
n is the index
n
k
is the principal nth root of k
5
5
25 = 32
32  2
(-2)5 = -32
625  5
54 = 625
4
4
32  2
625  not real
Index of n
index
radical sign
n
nth Root
Index of n
a
radicand
of a number is one of n
equal factors whose product is
that number
has an index where n is any
counting number n k
principal odd roots
5
32  2
5
32  2
principal even roots
6
64  2
6
64  not real
Radical Rules!
True or False:
64  16  4
64  64
88
T
25 4  25  4
100  5  2
10  10
T
50  25  2  25  2
 5 2
T
Radical Rule #1
In general, for non-negative numbers a, b and n
ab  a  b
n
a bna
n
b
simplified
Example:
 3 2  6
36  9  4  9  4
3
2
5
4
2

x

x

x
x

x

x

x
x  x
2
3
x5  x3  x5  x3
3
x7
3
 x
x 2  3 x7 x 2  3 x9
8
= x4
= x3
Hint: will the index divide evenly
into the exponent of radicand term?
Radical Rule #2
True or False:
If
25 5
4 
2
b
b
then
25
4 
b
and
T
25
b4
5
2 
b
25
b4
T
Transitive Property
of Equality
If a = b, and b = c,
then a = c
In general, for non-negative numbers a, b, and n
a

b
Example:
a
b
n
n
a
a
 n
b
b
4
144

81
12
144

 113
81
9
625 
256
4
4
625
5

4
256
Perfect Squares – Index 2
12
144
121
100
81
11
10
9
8
64
7
49
6
36
5
25
4
16
3
9
2
1
4
1
1
2
3
4
5
6
7
8
9 10 11 12
Perfect Square Factors
Find as many combinations of 2 factors
whose product is 75
1  75
3  25
5  15
Find as many combinations of 2 factors
whose product is 128
2  64
3  32
8  16
Factors
that are
Perfect
Squares
Simplifying Radicals
Simplify:
80
Find as many
combinations of 2
factors whose product
is 80
1 80
2 40
4 20
5 16
8 10
answer must be in radical form.
80
5  16
ab  a  b
perfect
square
16  4
comes out
4 5 from under
the 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 =
53 =
63 =
73 =
(x4)3 =
(-2y2)3 =
64
125
216
343
x12
-8y6
Simplifying Radicals
Simplify:
3
48
answer must be in radical form.
1) Factor the radicand
so that the perfect
power (cube) is a factor
2) Express the radical
as the product of the
roots of the factors
3) Simplify the radical
containing the largest
perfect power (cube)
 386
3
48
3
48  3 8
3
48
3
6
 23 6
Simplifying Radicals
3
Simplify:
4
3
1) Change the radicand
to an equivalent fraction
whose denominator is a
perfect power.
2) Express the radical as
the quotient of two roots
3) Simplify the radical in
the denominator
3
3
3
4
3

4
3
3 2
4 2

3
6
8

3
6
3
8
3
6

2
Model Problems
Simplify:
KEY: Find 2 factors - one
of which is the largest
perfect square possible
20
5 4
5  4  52  2 5
2 2
96
96
16 6 4 6
2 6
6




2
12
12
43 2 3
3
3
4 12  4 4  3  4 4  3  4  2 3  8 3
1
2
80  1
16  5  1
16  5
2
2
 1 4 5  2 5
2
Model Problems
Simplify:
4 3
45a b
 9 5(a 2 )2 b 2 b
 9
2 2
(a )
b
 3a b 5b
2
3
16x
6
 8 2( x )
2 3
3
 38
3
2
 2x
23
2
3
( x 2 )3
2
5
b