Download Properties of Exponents

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Compare the following algebraic expressions:
1. 3x  4 x  2x
2.
3.
4.
3(6x)
4(x  y)
0 x
and
and
and
and
x x x
3
4
6 3
(x )
4
(xy)
x
0
2
Aim: How do we Interpret and
Work with Exponents?
What is a “power”?
Ax
Powers
sometimes have
coefficients
BASE
b
EXPONENT
3x  4 x  2x
and
(3 4  2)x
9x
When adding terms,
we combined the
COEFFICIENTS.
x x x
3
4
x
34 2
x

2
9
When multiplying terms,
we combined the
EXPONENTS.
Product of a Powers Property
• When multiplying powers WITH THE
SAME BASE, you add their exponents.
x x x
a
In general:
b
a b
Examples:
4 y 3y 12y

6
7
2s 3t s  6s t
6
2
3
9 2
3(6x)
6x + 6x +6x
(6+6+6)x
18x
Here, the two
coefficients were
multiplied

6 3
and
(x )
x x x 
6
6
x
6
666
x
18
Here, the two
exponents were
multiplied.
Power of a Power Property
• When a power is being raised to a
power, you can simplify the expression
by multiplying the two exponents.
In general:
x 
m n
x
m n
Examples:
a   a
7 2
14
3   3
q r
qr
4(x  y)
and
4x  4y
(xy)
4
(xy)(xy)( xy)(xy)
x x x x y y y y
(commutative property of multiplication)

 the
Here, we distributed
coefficient to both parts in the
parentheses.
4
x y
4
Here, we distributed the
exponent to both parts in the
parentheses.
Power of a Product Property
• When a product has an exponent, that
exponent is applied to all parts of the
product.
In general:
(xy)  x y
a
a
a
Examples:
(3c)  3
7
7
c

7
(2xy z)  2 x y z
2
3
3
3
6 3
0 x
and
x
x 1
0
0
But WHYYYYYYY?!
02
0
2
2
x x x
So,


0


x
x x x
0
2
2
Must
be the multiplicative
identity: 1
Zero Power Property
• ANYTHING (besides 0) raised to the 0
power is equal to 1.
In general:

a  1, a  0
0