Which of the following is equivalent to $\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}}$? What is $\sqrt[3]{\frac{16 y^4}{x^6}}$ in simplest form, where $x \geq 0$ and $y \geq 0$?
A. $\sqrt[3]{16 x^6 y^4}$
B. $\frac{4 y^2}{x^3}$
C. $\sqrt[3]{\frac{y^4}{16 x^6}}$ $\frac{2 y(\sqrt[3]{2 y})}{x^2}$
D. $\sqrt[3]{\frac{16 y^4}{x^6}}$ $\frac{8 y(\sqrt[3]{2 y})}{v^3}$
Answer (2)
The simplified expression for both parts equals x 2 2 y ( 3 2 y ) , matching option C. Hence, the equivalent option is option C. The simplifications involve manipulating cube roots and algebraic fractions step-by-step.
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Simplify the first expression: 3 2 x 9 y 2 3 32 x 3 y 6 = 3 x 6 16 y 4 .
Simplify the second expression: 3 x 6 16 y 4 = x 2 3 16 y 4 .
Further simplify: 3 16 y 4 = 2 y 3 2 y .
Combine the terms to get the final simplified expression: x 2 2 y 3 2 y . The answer is x 2 2 y ( 3 2 y ) .
Explanation
Problem Analysis We are given two expressions to simplify and compare with the provided options. The expressions are 3 2 x 9 y 2 3 32 x 3 y 6 and 3 x 6 16 y 4 , where x ≥ 0 and y ≥ 0 .
Simplifying the First Expression First, let's simplify the expression 3 2 x 9 y 2 3 32 x 3 y 6 . We can combine the cube roots into a single cube root: 3 2 x 9 y 2 3 32 x 3 y 6 = 3 2 x 9 y 2 32 x 3 y 6 = 3 x 6 16 y 4 So, the first expression simplifies to 3 x 6 16 y 4 .
Simplifying the Second Expression Now, let's simplify the expression 3 x 6 16 y 4 . We can rewrite this as: 3 x 6 16 y 4 = 3 x 6 3 16 y 4
Simplifying the Cube Root of 16 We can further simplify the cube root of 16 as follows: 3 16 = 3 8 ⋅ 2 = 3 2 3 ⋅ 2 = 2 3 2
Simplifying the Cube Root of y^4 We can further simplify the cube root of y 4 as follows: 3 y 4 = 3 y 3 ⋅ y = y 3 y
Simplifying the Cube Root of x^6 We can further simplify the cube root of x 6 as follows: 3 x 6 = x 2
Combining Simplified Terms Combining these simplified terms, we get: 3 x 6 3 16 y 4 = x 2 2 3 2 ⋅ y 3 y = x 2 2 y 3 2 y
Final Answer Therefore, the simplified form of both expressions is x 2 2 y 3 2 y . Comparing this with the given options, we find that the equivalent expression is x 2 2 y ( 3 2 y ) .
Examples
Imagine you are designing storage containers, and you need to optimize the dimensions to hold a specific volume of liquid. Simplifying radical expressions helps you determine the exact measurements required, ensuring efficient use of space and resources. This is also applicable in fields like architecture, where precise calculations are essential for structural integrity and aesthetic design.
