Which of the following is equivalent to $\frac{\sqrt[3]{32 x^3 y^6}}{\sqrt[3]{2 x^9 y^2}}$ , where $x \geq 0$ and $y \geq 0$ ?
A. $\sqrt[3]{16 x^6 y^4}$
B. $\sqrt[3]{\frac{y^4}{16 x^6}}$
C. $\sqrt[3]{\frac{16 y^4}{x^6}}$
Answer (2)
The equivalent expression to 3 2 x 9 y 2 3 32 x 3 y 6 is 3 x 6 16 y 4 , which corresponds to Option C from the given choices.
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Combine the cube roots: 3 2 x 9 y 2 3 32 x 3 y 6 = 3 2 x 9 y 2 32 x 3 y 6 .
Simplify the fraction inside the cube root: 3 2 x 9 y 2 32 x 3 y 6 = 3 x 6 16 y 4 .
Compare the simplified expression with the given options.
The equivalent expression is 3 x 6 16 y 4 .
Explanation
Understanding the Problem We are given the expression 3 2 x 9 y 2 3 32 x 3 y 6 with the conditions x ≥ 0 and y ≥ 0 . Our goal is to simplify this expression and determine which of the provided options is equivalent.
Simplifying the Expression First, 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 Now, we simplify the fraction inside the cube root: 3 2 x 9 y 2 32 x 3 y 6 = 3 x 6 16 y 4 Thus, the simplified expression is 3 x 6 16 y 4 .
Comparing with Options Comparing the simplified expression with the given options: Option 1: 3 16 x 6 y 4 Option 2: 3 16 x 6 y 4 Option 3: 3 x 6 16 y 4 We see that Option 3 matches our simplified expression.
Final Answer Therefore, the equivalent expression is 3 x 6 16 y 4 .
Examples
Cube roots are used in various fields such as engineering and physics to calculate volumes and dimensions. For example, if you have a cube-shaped container and you know its volume, you can use the cube root to find the length of one side. Suppose a cube has a volume of V . Then the side length s is given by s = 3 V . This concept is also used in acoustics to determine the resonant frequencies of enclosed spaces, and in financial calculations involving growth rates.
