Digital Lesson
Exponents and
Radicals
Repeated multiplication can be written in
exponential form.
aaaaaa  a
6
3y 3y 3y   (3y)
34
3
exponent
base
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2
For all real numbers a and b and all integers m and n,
mn
1. a a  a
33  32  33 2  35  243
m n
m
mn
a
2.

a
an
x12  x125  x 7
x5
n
3. a  1n 
a
z 3  13 
z
 

1
a
1
z
n
3
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3
For all real numbers a and b and all integers m and n,
4. a0  1, a  0
1000  1
5. (ab)m  a mbm
(3x)4  34 x 4  81x 4
6. (a )  a
( y 2)6  y 2(6)  y12
m n
mn
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4
For all real numbers a and b and all integers m and n,
7.
 
 
a
b
d
5
m
m
a
 m
b
2
2
2
d
d
 2 
25
5
8. a  a  a 2
2
2
(4)  4  42  16
2
2
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5
Scientific notation is a convenient way to write
very large or very small numbers.
c 10
n
integer
1  c  10
Example:
a.) -3.683 x 107 = -36830000.
7 decimal places
b.) 0.495 x 10-5 = 0.00000495
5 decimal places
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6
A square root of a number is one of its two equal
factors.
25  5  5  5
2
A cube root of a number is one of its three equal
factors.
64  (4)(4)(4)  (4)
ab
3
n
nth root of a
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7
The principal nth root of a is the nth root that has
the same sign as a.
index
radical
symbol
n
a
radicand
4
81 =3
 3 125 =  5
3 is the principal 4th root of 81.
5 is the principal cube root of 125.
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8
For all real numbers a and b and all positive
integers m and n,
1.
2.
3.
 a
  64 
a 
n
m
2
  4   16
2
3
64
n
a  n b  n ab
3
8  3 27  3 216  6
n
2
m
n
a n
n
b
16 
8
3
a, b  0
b
16  2
8
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9
For all real numbers a and b and all positive
integers m and n,
4.
m n
3 4
5.
24  12 24
 a  a
 60   60
n
5
6.
a  mn a
n
5
If n is even,
n
a  a.
4
(36)4  36  36
If n is odd,
n
a  a.
5
(36)  36
n
n
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5
10
Example: Rationalize the denominator.
2  2  5 2 5
5
5
5 5
a.)
rationalizing
factor
5  14  2
5

14  2
14  2 14  2
b.)

5

conjugate

14  2
 5 14  10  14  2
10
2
14  4
simplified
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11
rational
exponent
64
23
 64
3
2
Example: Simplify the radical expression.
a.)
b.)
4
24
12

3

3
 3
3
2
32  4 32  4 16  2  2 4 2
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12
An expression involving radicals is in simplest
form when the following are satisfied.
1. All possible factors have been removed from the radical.
24  4  6  2 6
2. The denominators have been rationalized.
3
 3  5 3 5 3 5
7 5 7 5 5 7(5) 35
3. The index of the radical is reduced.
3
3
36
12
125  6 125  6 (5)  5  5  5
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13