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# 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

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 Â· 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 Â· 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.