So I've been pondering this question for a while now and I cant see anything wrong. Please tell me where I go wrong in the following equations:

If...

\(\sqrt{ab^2}=b\sqrt{a}\)

Then

\(\sqrt{5}=\sqrt{5(-1)^2}=(-1)\sqrt5\)

Therefore

\(\sqrt5=-\sqrt5\)

How is this possible. (There has to be some simple detail I missed)

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## Comments

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TopNewestRemember that \(\sqrt{x^{2}} \not = x \Rightarrow \sqrt{x^{2}} = |x|\)

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The contradiction you get is precisely why mathematicians decided to say \( \sqrt{x^{2}} = |x|\)

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I'll work with this case if necessary \( \sqrt{ab^{2}} = b\sqrt{a}\) if b > 0 and \( \sqrt{ab^{2}} = -b\sqrt{a}\) if b < 0. Since in this case \(b = -1 \rightarrow \sqrt{5}= \sqrt{5*(-1)^{2}} = -(-1)\sqrt{5} = \sqrt{5}\)

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