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Re: Which of the following expressions is equivalent to [square_root]x
[#permalink]
28 Oct 2021, 05:35
1
Expert Reply
GreenlightTestPrep wrote:
Which of the following expressions is equivalent to \(\sqrt{x^{16}}\) for all non-zero values of \(x\)?
A) \(x^{-4}\)
B) \(-x^{-4}\)
C) \(-x^{4}\)
D) \(x^{4}\)
E) \(x^{8}\)
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 \(k=8\)