What is the product of the monomials given?
-18(ab)^{13}
-18a^7b^6
-18a^{10}b^6
-18a^7b^9
Answer (2)
Multiply the coefficients: ( − 18 ) × ( − 18 ) × ( − 18 ) × ( − 18 ) = 104976 .
Multiply the variables: ( ab ) 13 × a 7 b 6 × a 10 b 6 × a 7 b 9 = a 37 b 34 .
Combine the results: 104976 a 37 b 34 .
The product of the monomials is 104976 a 37 b 34 .
Explanation
Understanding the Problem We are asked to find the product of four monomials: − 18 ( ab ) 13 , − 18 a 7 b 6 , − 18 a 10 b 6 , and − 18 a 7 b 9 . To do this, we will multiply their coefficients and then multiply their variable parts, using the rules of exponents.
Multiplying the Coefficients First, let's multiply the coefficients: ( − 18 ) × ( − 18 ) × ( − 18 ) × ( − 18 ) = ( − 18 ) 4 The result of this calculation is 104976.
Multiplying the Variables Next, we multiply the variable parts. We have ( ab ) 13 × a 7 b 6 × a 10 b 6 × a 7 b 9 . Using the exponent rule x m × x n = x m + n , we can simplify this expression. First, we rewrite ( ab ) 13 as a 13 b 13 . Then we have: a 13 b 13 × a 7 b 6 × a 10 b 6 × a 7 b 9 = a 13 + 7 + 10 + 7 b 13 + 6 + 6 + 9 = a 37 b 34 .
Combining the Results Finally, we combine the results from the coefficient and variable parts to get the final product: 104976 a 37 b 34 .
Examples
Understanding monomials and their products is essential in various fields, such as physics and engineering, where complex equations often involve polynomial expressions. For example, when calculating the volume of a complex shape or modeling the trajectory of a projectile, you might encounter expressions involving monomials. By simplifying and manipulating these expressions, engineers and scientists can make accurate predictions and design efficient systems. This skill is also crucial in computer graphics, where monomials are used to define curves and surfaces.
The product of the given monomials is 104976 a 37 b 34 . This result comes from multiplying the coefficients and summing the exponents of the variables accordingly. So, you have a final result combining coefficients with a combined variable expression.
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