Sect 5.1 - Exponents: Multiplying And Dividing Common Bases Worksheet With Answers Page 3
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Multiplication of Like Bases (The Product Rule for Exponents)
If m and n are positive integers and a is a non-zero real number,
m
• • • • a
n
m + n
then a
= a
.
Property #1
In words, when multiplying powers of the same base, add the
exponents and keep the same base.
Simplify the following:
7
• y
8
11
• (– 5y)
15
Ex. 5a
y
Ex. 5b
(– 5y)
(– 5y)
14
• 7
12
2
• a
5
• a • b
3
• b • b
11
Ex. 5c
7
Ex. 5d
a
Solution:
7
• y
8
7 + 8
15
a)
y
= y
= y
.
11
• (– 5y)
15
11 + 15 + 1
27
b)
(– 5y)
(– 5y) = (– 5y)
= (– 5y)
.
14
• 7
12
14 + 12
26
c)
7
= 7
= 7
.
2
• a
5
• a • b
3
• b • b
11
2 + 5 + 1
3 + 1 + 11
8
15
d)
a
= a
b
= a
b
8
15
Notice that we cannot simplify a
b
since the bases are not the same.
Keep in mind that property #1 only works for multiplication involving one
term. It does not work for addition and subtraction.
4
• 5x
4
4
4
Ex. 6a
3x
Ex. 6b
3x
+ 5x
3
4
3
4
Ex. 6c
x
– x
Ex. 6d
x
(– x
)
3
2
Ex. 6e
(– 4x
)(– 7y
)
Solution:
4
• 5x
4
= 3 • 5x
4
• x
4
8
a)
3x
= 15x
.
b)
Caution, the operation is addition. Combine like terms:
4
4
4
3x
+ 5x
= 8x
.
3
4
c)
There are no like terms to combine, so our answer is x
– x
.
3
4
3 + 4
7
d)
x
(– x
) = – x
= – x
3
2
3
2
3
2
e)
(– 4x
)(– 7y
) = 28x
y
. We cannot simplify the x
y
since the
bases are not the same.
8
y
Ex. 7
5
y
Solution:
Write both the numerator and denominator in expanded form and
divide out the common factors of y:
8
y•y •y•y •y•y •y•y
y•y •y•y •y•y •y•y
y•y •y•1•1•1•1•1
y
3
=
=
=
= y
.
y •y•y •y•y
y •y•y •y•y
5
1•1•1•1•1
y
This example illustrates our second property:
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