To simplify the expression \( \sqrt[6]{x^{30}} \), we can rewrite it using exponent notation.
The general rule for radicals is that:
\[ \sqrt[n]{a^m} = a^{\frac{m}{n}} \]
Applying this rule to our expression:
\[ \sqrt[6]{x^{30}} = x^{\frac{30}{6}} = x^5 \]
Now, we can examine the multiple choice options provided:
- \( x^5 \sqrt[6]{x^3} \)
- \( x^4 \sqrt[6]{x^3} \)
- \( x^{5x} \)
- \( x^{4x} \)
Since we simplified \( \sqrt[6]{x^{30}} \) down to \( x^5 \), we are looking for the equivalent representation or any combination of \( x^5 \) that directly can be shown as one of the options.
Given the options:
- The expression \( x^5 \sqrt[6]{x^3} \) can be rewritten as \( x^5 \cdot x^{\frac{3}{6}} = x^5 \cdot x^{0.5} = x^{5.5} \) which does not match.
- The expression \( x^4 \sqrt[6]{x^3} \) becomes \( x^4 \cdot x^{0.5} = x^{4.5} \), which also does not match.
- The expressions \( x^{5x} \) and \( x^{4x} \) do not relate to the simplified form since they suggest multiplication of a power rather than simplification.
From the options, \( x^5 \sqrt[6]{x^3} \) seems to best relate to the base form \( x^5 \) because it adds to it, but if looking for equivalency directly, \( x^5 \) by itself is accurate.
Thus, the correct simplification of the given expression \( \sqrt[6]{x^{30}} \) is:
Answer: \( x^5 \)
However, since that exact answer is not listed among your options explicitly, the closest would be the one implying \( x^5 \) itself or engagement with different forms.
Overall, the corresponding multiple-choice answer directly representing our simplification is \( x^5 \).