To simplify the expression \( x^{16} \cdot x^{16} \) using the properties of exponents, we can apply the rule that states \( a^m \cdot a^n = a^{m+n} \).
Thus, we have:
\[ x^{16} \cdot x^{16} = x^{16+16} = x^{32} \]
Now we need to express \( x^{32} \) in terms of roots. The expression \( x^{32} \) can be related to roots as follows:
\[ x^{32} = \left(x^{1/2}\right)^{64} = \sqrt{x}^{64} \]
However, we need to match it to one of the given options, so let's express it as a root directly:
- The 36th root of \( x \) is \( x^{1/36} \).
- The cube root of \( x \) is \( x^{1/3} \).
- The square root of \( x \) is \( x^{1/2} \).
- \( \frac{1}{x^{\sqrt{3}}} = x^{-\sqrt{3}} \).
None of these expressions directly give us \( x^{32} \). Hence, we should also consider reciprocal forms if relevant.
The correct simplification for \( x^{32} \) does not seem to match any of the options provided, but if you were looking for an equivalent root expression for something like \( x^{1/1} \) (i.e., \( x \)) it would lead to adjustments.
However, since none of the forms given clearly correspond to \( x^{32} \), I suspect there may be a misunderstanding or a mislabeling in the options. Would you please confirm the representations to ensure we are matching the right expressions?
If you just need the simplified answer based solely on the options, \( \sqrt{x} \) is \( x^{1/2} \) and does not relate, but it appears the options may have limitations or errors in their approach.
If you need any follow-up based on specific content or properties of these roots in exponent discussion, please do ask!