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Multiplying MonomialsAfter studying this lesson, you will be able to:
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} 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.
