Here are two approaches....

I created this question to highlight a common mistake some students make when it comes to finding the square root of a power.
These students incorrectly assume that, since \(\sqrt{16}=4\), the square root of \(x^{16}\) must be \(x^4\).
However, a quick test shows that this is not the case.
If \(\sqrt{x^{16}}=x^4\), then it must be true that \((x^4)(x^4) = x^{16}\). This, of course, is incorrect since \((x^4)(x^4) = x^{8}\)

Solution #1: Apply the property that says \(\sqrt{k} = k^{\frac{1}{2}}\)
This means we can rewrite the given expression as follows: \(\sqrt{x^{16}} = (x^{16})^{\frac{1}{2}}\)

Now apply the power of a power law to get: \(\sqrt{x^{16}} = x^8\)

Answer: E


Solution #2: Let \(x^k = \sqrt{x^{16}}\)

This means we can write: \((x^k)(x^k) = x^{16}\)

Now apply the product law to get: \(x^{2k} = x^{16}\)

Since the bases are equal we know that \(2k = 16\), which means \(k=8\)

In other words, \(x^8 = \sqrt{x^{16}}\)

Answer: E

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