Multiplying Monomials
After studying this lesson, you will be able to:
- Multiply monomials.
- Multiply powers with the same base.
- Raise a power to a power.
Monomials have one term. The term can be a
number, a variable, or the product of a number and a variable.
Monomials are expressions with do not contain a + or - sign....
it has only one term.
Multiplying Powers with the Same Base:
The base stays the same; Add exponents
We are now ready for the next exponent rule:
Power of a Power: Raise the coefficient to
the power; Multiply the exponents The power of a power rule is
used when we are raising one power to another power.
Example 1
( x 2 ) 3
In this problem we have x 2 raised to the 3 rd
power. Our coefficient is 1. Since we have parentheses,
everything in the parentheses must be raised to the power of 3.
If we raise the coefficient (1) to the 3 rd power, we have 1.
Using the power of a power rule, we multiply the exponents 2
times 3.
Therefore, our answer is 1x 6 or x 6
Example 2
( x y 2 z 3 ) 2
In this problem we have x y 2 z 3 raised
to the 2 nd power. Our coefficient is 1. Since we have
parentheses, everything in the parentheses must be raised to the
power of 2. If we raise the coefficient (1) to the 2 nd power, we
have 1. Using the power of a power rule, we multiply the all the
exponents in the parentheses by 3.
Therefore, our answer is x 2 y 4 z
6
Example 3
( 2x ) 3
In this problem we have 2x raised to the 3 rd
power. This time our coefficient is 2. Since we have parentheses,
everything in the parentheses must be raised to the power of 3.
If we raise the coefficient (2) to the 3 rd power, we
have 8 because 2 times 2 times 2 is eight. Using the power of a
power rule, we multiply the exponents 1 times 3.
Therefore, our answer is 8x 3
Example 4
( -4 x y 2 ) 2
In this problem we have -4 x y 2 raised to the 2 nd
power. Our coefficient is -4. Since we have parentheses,
everything in the parentheses must be raised to the power of 2.
If we raise the coefficient (-4) to the 2 nd power, we
have 16 because -4 times -4 is sixteen. Using the power of a
power rule, we multiply the exponents 1 and 2 by 2.
Therefore, our answer is 16 x 2 y 4
For the next group of problems, we will combine what we've
learned in this section with the order of operations. Remember to
use the correct order of operations:
Parentheses
Exponents
Multiplication
Division
Addition
Subtraction
Example 5
( 2x 2 ) 3 ( 3x 4 )
In this problem we have 2x 2 being raised to the3
rd power. Then we will need to multiply that answer by 3x
4 . We do the first parentheses first because it is being
raised to a power. Following the order of operations, we always
do the exponents before we multiply. 2x 2 being raised
to the 3 rd power will give us 8x 6.
Next, we will multiply 8x 6 · 3x 4
Remember the rules for multiplying. We multiply the
coefficients (8 and 3) and we add the exponents (6 and 4).
This will give us the answer: 24x 10
Example 6
( 6ab 2 ) 3 ( 5a ) 2
In this problem we have 6ab 2 being raised to the 3
rd power and we have 5a being raised to the 2 nd
power. We do the first parentheses first because both are being
raised to a power. Following the order of operations, we always
do the exponents before we multiply. 6ab 2 being
raised to the 3 rd power will give us 216 a 3 b
6 . 5a being raised to the 2 nd power will give
us 25 a 2
Next, we will multiply 216 a 3 b 6 · 25
a 2 to giveus 5400 a 5 b 6 Remember
the rules for multiplying. We multiply the coefficients (216 and
25) and we add the exponents.
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