Which expression is equivalent to $\sqrt[3]{\frac{10 x^5}{54 x^8}}$ ? Assume $x \neq 0$
A. $\frac{\sqrt[3]{10 x}}{3 x^2}$
B. $\frac{x(\sqrt[3]{5 x})}{3}$
C. $\frac{3(\sqrt[3]{5 x})}{x}$
D. $\frac{\sqrt[3]{5}}{3 x}$
Answer (2)
Simplify the fraction inside the cube root: 54 x 8 10 x 5 = 27 x 3 5 .
Rewrite the expression: 3 27 x 3 5 .
Apply the cube root to numerator and denominator: 3 27 x 3 3 5 .
Simplify the denominator: 3 x 3 5 .
The equivalent expression is 3 x 3 5 .
Explanation
Understanding the Problem We are given the expression 3 54 x 8 10 x 5 and we want to find an equivalent expression. We are also given that x = 0 .
Simplifying the Fraction First, let's simplify the fraction inside the cube root. We have 54 x 8 10 x 5 = 54 10 ⋅ x 8 x 5 = 27 5 ⋅ x 5 − 8 = 27 5 ⋅ x − 3 = 27 x 3 5 .
Rewriting the Expression Now, we can rewrite the original expression as 3 54 x 8 10 x 5 = 3 27 x 3 5 .
Applying the Cube Root Next, we apply the cube root to both the numerator and the denominator: 3 27 x 3 5 = 3 27 x 3 3 5 .
Simplifying the Denominator We simplify the cube root in the denominator. Since 27 = 3 3 , we have 3 27 = 3 . Also, 3 x 3 = x . Therefore, 3 27 x 3 = 3 27 ⋅ 3 x 3 = 3 x .
Final Simplification So, the expression becomes 3 27 x 3 3 5 = 3 x 3 5 .
Finding the Equivalent Expression Comparing this with the given choices, we see that the equivalent expression is 3 x 3 5 .
Examples
Imagine you are designing a container, and its volume is given by the expression 3 54 x 8 10 x 5 . Simplifying this expression allows you to easily determine how the volume changes with respect to x , which could represent a dimension of the container. This type of simplification is useful in engineering and design to optimize shapes and sizes of objects.
The expression 3 54 x 8 10 x 5 simplifies to 3 x 3 5 . The option that corresponds to this simplified expression is option D: 3 x 3 5 .
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