Multiply.
[tex]$2 v^3 \cdot 2 w^7 v^5 \cdot 4 w$[/tex]
Simplify your answer as much as possible.
Answer (1)
Multiply the coefficients: 2 c d o t 2 c d o t 4 = 16 .
Combine the 'v' terms: v 3 c d o t v 5 = v 3 + 5 = v 8 .
Combine the 'w' terms: w 7 c d o tw = w 7 + 1 = w 8 .
The simplified expression is 16 v 8 w 8 .
Explanation
Understanding the problem We are asked to multiply and simplify the expression 2 v 3 c d o t 2 w 7 v 5 c d o t 4 w . This involves combining like terms by adding their exponents.
Multiplying the coefficients First, we multiply the coefficients: 2 c d o t 2 c d o t 4 = 16 .
Combining 'v' terms Next, we combine the terms with the variable 'v': v 3 c d o t v 5 = v 3 + 5 = v 8 . Remember that when multiplying terms with the same base, we add the exponents.
Combining 'w' terms Then, we combine the terms with the variable 'w': w 7 c d o tw = w 7 + 1 = w 8 . Here, w is the same as w 1 , so we add the exponents 7 and 1.
Writing the final expression Finally, we write the simplified expression as the product of the coefficient and the combined variable terms: 16 v 8 w 8 .
Final Answer Therefore, the simplified expression is 16 v 8 w 8 .
Examples
Imagine you're calculating the volume of a rectangular prism where the length involves variable v and the width involves variable w . If the length is 2 v 3 , the width is 2 w 7 v 5 , and the height is 4 w , then the volume is the product of these three expressions. Simplifying this product, as we did in the problem, gives you a concise formula for the volume in terms of v and w . This kind of simplification is useful in various engineering and physics problems where you need to express complex quantities in a more manageable form.
