Question
Which is equivalent to \( \sqrt[5]{1,215}^{x} \) ? \( 243^{x} \) \( 1,215^{\frac{1}{5} x} \) \( 1,215^{\frac{1}{5 x}} \) \( 243^{\frac{1}{x}} \)
Ask by Mann Lawson. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( 1,215^{\frac{1}{5} x} \)
Solution
The expression \( \sqrt[5]{1,215}^{x} \) can be rewritten using exponents as follows:
\[
\sqrt[5]{1,215}^{x} = (1,215)^{\frac{1}{5}}^{x} = 1,215^{\frac{1}{5} \cdot x}
\]
This simplifies to:
\[
1,215^{\frac{1}{5}x}
\]
So, among the given options, the equivalent expression is:
**\( 1,215^{\frac{1}{5} x} \)**
**Answer:** \( 1,215^{\frac{1}{5} x} \)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To determine which expression is equivalent to \( \sqrt[5]{1,215}^{x} \), we start by rewriting the expression. The fifth root can be expressed as an exponent: \[ \sqrt[5]{1,215} = 1,215^{\frac{1}{5}}. \] Now, raising this to the power \( x \) gives us: \[ \sqrt[5]{1,215}^{x} = \left(1,215^{\frac{1}{5}}\right)^{x} = 1,215^{\frac{x}{5}}. \] Now, let's analyze the provided options. The second option, \( 1,215^{\frac{1}{5} x} \), is the same as \( 1,215^{\frac{x}{5}} \), confirming that it is indeed equivalent. Thus, the correct answer is: \( 1,215^{\frac{1}{5} x} \).
