# How to simplify an expression like this: $(x^2+x^{-2}-2)^{1/2}$

Sorry, I am not sure how to do the maths mark-up on this site but hopefully the question will make sense. I should know how to do this, but I have got myself stuck! Can anyone help?

$(x^2+x^{-2}-2)^{1/2}$

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## 4 Answers

$$\left(x-\frac{1}{x}\right)^2= \dots?$$

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Thanks! Took me a moment to figure out what you were on about, but I do see how to simplify it now. So the smallest value I get factorising to get that and then substituting it in should be x-(1/x)... Assuming I understood you correctly, that is... – Magpie Jul 26 '12 at 16:55
+1 for the question mark. – dot dot Jul 26 '12 at 17:03
@Magpie, I'm not sure what you mean by "the smalles value I get factorising...". The fact is your expression $\,x^2-2+x^{-2}\,$ has the expected form for the well-known squared binomial expression: (term 1 squared) + (term 2 squared) $\,\pm\,$ (twice term 1 times term 2) = (term 1 $\,\pm\,$ term 2)^2...practice, that's all. – DonAntonio Jul 26 '12 at 18:36
@DonAntonio Your answer is contradictory to jasoncube's. The expression is factored appropriately, however the initial exponent of $1/2$ should cancel out the exponent of $2$. Am I misunderstanding something? – user26649 Jul 27 '12 at 12:16
@FarhadYusufali, I think you are. I can't see how my answer is "contradictory to jasoncube's". IMO, both are accurate – DonAntonio Jul 27 '12 at 12:50

$$\sqrt{x^2 + x^{-2} - 2} = \sqrt{x^2 – 2(x)(x^{-1})+ (x^{-1})^2} =\sqrt{(x – x^{-1})^2} = |x – x^{-1}|$$

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What is $x^{-2}$ equal to? Hint: can you write it as a fraction? If so, I would then look at adding and subtracting fractions and go from there.

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For $x\ne 0$, $x^2+x^{-2}-2=(x^4-2x^2+1)/x^2=(x^2-1)^2/x^2$. So taking square roots of both sides we get on the right side $|(x^2-1)/x|=|x-1/x|$.

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