To determine which expression is equivalent to [tex]\(\sqrt[6]{x^5}\)[/tex], we need to rewrite the given expression using the properties of exponents.
The 6th root of [tex]\(x^5\)[/tex] can be rewritten as an expression with a fractional exponent. The general property we use is:
[tex]\[
\sqrt[n]{a} = a^{\frac{1}{n}}
\][/tex]
Therefore:
[tex]\[
\sqrt[6]{x^5} = (x^5)^{\frac{1}{6}}
\][/tex]
When we have a power raised to another power, we multiply the exponents:
[tex]\[
(x^5)^{\frac{1}{6}} = x^{5 \times \frac{1}{6}} = x^{\frac{5}{6}}
\][/tex]
Thus, the expression [tex]\(\sqrt[6]{x^5}\)[/tex] simplifies to [tex]\(x^{\frac{5}{6}}\)[/tex].
Comparing this result with the given choices:
- A: [tex]\(\frac{x^5}{6}\)[/tex]
- B: [tex]\(x^{\frac{5}{6}}\)[/tex]
- C: [tex]\(\frac{6}{x^5}\)[/tex]
- D: [tex]\(x^{\frac{6}{5}}\)[/tex]
The correct answer is:
B. [tex]\(x^{\frac{5}{6}}\)[/tex]
