Which expression is equivalent to $\sqrt[3]{32 x^8 y^{10}}$?
A. $4 x^2 y^3(\sqrt[3]{2 x^2 y})$
B. $2 x^4 y^5(\sqrt[3]{4})$
C. $2 x^2 y^3(\sqrt[3]{4 x^2 y})$
D. $4 x^4 y^5(\sqrt[3]{2})$
Answer (2)
Rewrite the expression as ( 32 x 8 y 10 ) 3 1 .
Factor 32 and separate the terms: ( 2 5 x 8 y 10 ) 3 1 = ( 2 3 ⋅ 2 2 ⋅ x 6 ⋅ x 2 ⋅ y 9 ⋅ y ) 3 1 .
Group perfect cubes: ( 2 3 x 6 y 9 ) 3 1 ( 2 2 x 2 y ) 3 1 .
Simplify to obtain the final answer: 2 x 2 y 3 3 4 x 2 y .
2 x 2 y 3 ( 3 4 x 2 y )
Explanation
Understanding the Problem We are given the expression 3 32 x 8 y 10 and asked to find an equivalent expression from the given choices. We need to simplify the given expression by factoring out perfect cubes.
Rewriting the Expression We can rewrite the expression as ( 32 x 8 y 10 ) 3 1 .
Factoring 32 Factor 32 into 2 5 , so we have ( 2 5 x 8 y 10 ) 3 1 .
Separating Terms Separate the terms as ( 2 3 ⋅ 2 2 ⋅ x 6 ⋅ x 2 ⋅ y 9 ⋅ y ) 3 1 .
Grouping Perfect Cubes Rewrite the expression as ( 2 3 x 6 y 9 ) 3 1 ( 2 2 x 2 y ) 3 1 .
Simplifying Simplify the first term to get 2 x 2 y 3 ( 2 2 x 2 y ) 3 1 .
Final Simplification Rewrite the expression as 2 x 2 y 3 3 4 x 2 y .
Identifying the Answer Comparing this to the given choices, we see that the equivalent expression is 2 x 2 y 3 ( 3 4 x 2 y ) .
Examples
Simplifying radical expressions is useful in various fields, such as engineering and physics, where complex calculations often involve radicals. For example, when calculating the volume of a complex shape or analyzing the behavior of waves, simplifying radicals can make the calculations more manageable. Imagine you are designing a container, and its volume is expressed as 3 32 x 8 y 10 . By simplifying this to 2 x 2 y 3 3 4 x 2 y , you can easily determine the amount of material needed for construction based on the dimensions x and y .
The expression 3 32 x 8 y 10 simplifies to 2 x 2 y 3 ( 3 4 x 2 y ) , which matches option C. Therefore, the equivalent expression is option C.
;
