Which expression is equivalent to [tex]$\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2 ?[/tex]
Answer (2)
Expand the second term: ( 3 x 5 y ) 2 = 9 x 10 y 2 .
Multiply the first term by the expanded second term: ( 4 x 3 y 5 ) ( 9 x 10 y 2 ) .
Simplify the expression by multiplying the coefficients and adding the exponents: 36 x 13 y 7 .
The equivalent expression is 36 x 13 y 7 .
Explanation
Understanding the Problem We are given the expression ( 4 x 3 y 5 ) ( 3 x 5 y ) 2 and asked to find an equivalent expression. We will simplify the given expression using the rules of exponents.
Expanding the Second Term First, we need to expand the second term in the expression. Recall that when raising a product to a power, we raise each factor to that power. So, we have ( 3 x 5 y ) 2 = 3 2 ( x 5 ) 2 y 2 = 9 x 5 × 2 y 2 = 9 x 10 y 2 .
Substituting Back Now, we substitute this back into the original expression: ( 4 x 3 y 5 ) ( 3 x 5 y ) 2 = ( 4 x 3 y 5 ) ( 9 x 10 y 2 ) .
Multiplying the Terms Next, we multiply the two terms together. When multiplying terms with the same base, we add their exponents: 4 x 3 y 5 ⋅ 9 x 10 y 2 = ( 4 ⋅ 9 ) ⋅ ( x 3 ⋅ x 10 ) ⋅ ( y 5 ⋅ y 2 ) = 36 x 3 + 10 y 5 + 2 = 36 x 13 y 7 .
Final Answer Therefore, the expression equivalent to ( 4 x 3 y 5 ) ( 3 x 5 y ) 2 is 36 x 13 y 7 .
Examples
Understanding how to simplify expressions with exponents is crucial in many fields, such as physics and engineering. For example, when calculating the volume of a complex shape or determining the energy of a system, you often encounter expressions with exponents. Simplifying these expressions correctly allows for accurate calculations and predictions. Imagine you are designing a satellite; accurately calculating the surface area, which involves exponential terms, is essential for thermal management and ensuring the satellite functions correctly in space.
The equivalent expression to ( 4 x 3 y 5 ) ( 3 x 5 y ) 2 is 36 x 13 y 7 . We obtain this by first squaring the second term, then multiplying the results while applying the rules of exponents. Hence, we arrive at the simplified expression of 36 x 13 y 7 .
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