How do you simplify $(1 - x^2)^(1/2) - x^2(1 - x^2)^(-3/2)$ ?

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Answer 1 · 2015-07-04T02:53:09

$((-x^2+x+1)(-x^2-x+1))/(1-x^2)^(3/2)$

$(1-x^2)^(1/2)-x^2(1-x^2)^(-3/2)$

We will use : $color(red)(a^(-n) = 1/a^n)$

$<=> (1-x^2)^(1/2)-x^2/(1-x^2)^(color(red)(+3/2))$

We want two fractions with the same denominator.

$<=> ((1-x^2)^(1/2)*color(green)((1-x^2)^(3/2)))/color(green)((1-x^2)^(3/2))-x^2/(1-x^2)^(+3/2)$

We will use : $color(red)(u^(a)*u^(b) = u^(a+b))$

$<=> (color(red)((1-x^2)^(2)))/(1-x^2)^(3/2)-x^2/(1-x^2)^(3/2)$

$<=> ((1-x^2)^(2)-x^2)/(1-x^2)^(3/2)$

We will use the following polynomial identity :

$color(blue)((a+b)(a-b)=a^2-b^2)$

$<=> color(blue)((1-x^2+x)(1-x^2-x))/(1-x^2)^(3/2)$

$<=> ((-x^2+x+1)(-x^2-x+1))/(1-x^2)^(3/2)$

We can't do better than this, and now you can easily (if you want) find the solution of $((-x^2+x+1)(-x^2-x+1))/(1-x^2)^(3/2) = 0$

How do you simplify (1 - x^2)^(1/2) - x^2(1 - x^2)^(-3/2)(1 - x^2)^(1/2) - x^2(1 - x^2)^(-3/2)?