Laws of Exponents Objective: TSW simplify powers. TSW simplify radicals. TSW develop a vocabulary associated with exponents. TSW use the laws of exponents to simplify.
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Laws of Exponents
Objective:TSW simplify powers.TSW simplify radicals.
TSW develop a vocabulary associated with exponents.
TSW use the laws of exponents to simplify.
Exponents
The lower number is called the base and the upper number is called the exponent.
The exponent tells how many times to multiply the base.
Exponents
73
exponent
base
power
1. Evaluate the following exponential expressions:
A. 42 = 4 x 4 = 16 B. 34 = 3 x 3 x 3 x 3 = 81 C. 23 = D. (-1) =7
Squares
To square a number, just multiply it by itself.
=
= 3 x 3 = 93 squared =
Perfect Squares
1² = 1 2² = 4 3² = 9 4² = 16 5² = 25 6² = 36 7² = 49
8² = 64 9² = 81 10² = 100 11² = 121 12² = 144 13² = 169
Square Roots
A square root goes the other direction. 3 squared is 9, so the square root of 9 is 3
3 9
Square Roots
11
4 2
9 3
16 4
25 5
36 6
749
864
981
10100
11121
12144
13691
Radicals
- The inverse operation of raising a number to a power.
For Example, if we use 2 as a factor with a power of 4, then we get 16. We can reverse this by finding the fourth root of 16 which is 2.
= 2164
Radicals
In this problem, the 16 is called the radicand, the 4 is the index, and the 2 is the root.
The symbol is known as the radical sign. If the index is not written, then it is understood to be 2.
The entire expression is known as a radical expression or just a radical.
Example
Simplify:
a) c)
b) d)
8127
16 8
34
3
Laws of Exponents
Whenever we have variables which contain exponents and have equal bases, we can do certain mathematical operations to them.
Those operations are called the “Laws of Exponents.”
Laws of Exponents
m
mm
mnnm
mmmnmnm
y
x
y
xxx
yxxyxxx
.4.3
.2.1
Laws of Exponents
mnn
m
nmn
m
xx
xthenmnifb
xx
xthennmifa
1,.5
,.5
Zero Exponents
A nonzero based raise to a zero exponent is equal to one
a0 = 1
Negative Exponents
a-n= (
1______
an )
A nonzero base raised to a negative exponent is the reciprocal of the base raised to the positive exponent.
A nonzero base raised to a negative exponent is the reciprocal of the base raised to the positive exponent.
Basic Examples
32 xx 32x 5x
34x 34x 12x
Basic Examples
3xy 33yx
3
y
x3
3
y
x
4
7
x
x
1
47x 3x
7
5
x
x 57
1
x 2
1
x
Basic Examples
Examples
5
p
u
74 xx
57y
3
9
x
x
1.
2.
3.
4.
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