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# I think I broke math

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)

Note by Trevor Arashiro
2 years, 4 months ago

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Remember that $$\sqrt{x^{2}} \not = x \Rightarrow \sqrt{x^{2}} = |x|$$ · 2 years, 4 months ago

The contradiction you get is precisely why mathematicians decided to say $$\sqrt{x^{2}} = |x|$$ · 2 years, 4 months ago

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 $$b = -1 \rightarrow \sqrt{5}= \sqrt{5*(-1)^{2}} = -(-1)\sqrt{5} = \sqrt{5}$$ · 2 years, 4 months ago