To simplify the expression \( x^{16} \cdot x^{16} \), we can use the properties of exponents. When multiplying two expressions with the same base, we add their exponents:
\[ x^{16} \cdot x^{16} = x^{16 + 16} = x^{32} \]
Now, we need to see which of the given options is equivalent to \( x^{32} \).
- \( \sqrt{x} \) can be written as \( x^{1/2} \)
- \( \sqrt[36]{x} \) can be written as \( x^{1/36} \)
- \( \sqrt[3]{x} \) can be written as \( x^{1/3} \)
- \( \frac{1}{\sqrt[3]{x}} \) can be written as \( x^{-1/3} \)
Since all the options are expressed in forms with exponents less than or equal to 1, we see that none of them is equivalent to \( x^{32} \).
Thus, it looks like there may have been some confusion in the response options provided, as none truly represents \( x^{32} \). However, it seems that they are looking for a representation simplified to fractional exponents.
If you need to express \( x^{32} \) differently using roots, it would be:
\[ \sqrt[1/32]{x^{32}} \text{ or } \sqrt[32]{x^{32}} \]
But those options are not provided. Given the responses listed, they might be evaluating your understanding of square roots, cube roots, or the idea of reciprocal roots rather than matching with \( x^{32} \).
Let me know if you need clarification or help with a different expression!