Question
Find the root. Assume that all variables represent nonnegative real numbers. \( \sqrt[5]{x^{30}} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The root is not a real number. B.
Ask by Hobbs Ford. in the United States
Mar 13,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The root is \(x^6\).
Solution
We begin with the expression
\[
\sqrt[5]{x^{30}}.
\]
Recall that taking the fifth root of a number is equivalent to raising it to the power of \( \frac{1}{5} \). Thus, we have
\[
\sqrt[5]{x^{30}} = \left( x^{30} \right)^{\frac{1}{5}}.
\]
By using the exponent rule \(\left(a^m\right)^n = a^{mn}\), we get:
\[
\left( x^{30} \right)^{\frac{1}{5}} = x^{30 \times \frac{1}{5}} = x^6.
\]
Since \(x\) is assumed to be a nonnegative real number, the expression \(x^6\) is also a nonnegative real number.
Thus, the correct choice is (if option B indicates \(x^6\)):
\[
\boxed{x^6}.
\]
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The Deep Dive
The expression \( \sqrt[5]{x^{30}} \) can be simplified using the property of exponents. Since \( \sqrt[5]{x^{30}} = x^{30/5} = x^6 \), and assuming \( x \) is a nonnegative real number, the root is simply \( x^6 \). Now, if you consider any nonnegative real number for \( x \), \( x^6 \) will also be nonnegative, maintaining the real number condition. Thus, the root \( x^{30/5} \) is indeed a real number. Feel free to plug in any value for \( x \) to see how it behaves—whether it’s 0, 1, or any positive number, the result will always be valid!