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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?

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

4 Answers 4

18

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

  • 0
    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...2012-07-26
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    @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.2012-07-26
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    @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?2012-07-27
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    @FarhadYusufali, I think you are. I can't see how my answer is "contradictory to jasoncube's". IMO, both are accurate2012-07-27
10

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

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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.

1

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 $|(x^2-1)/x|=|x-1/x|$.