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I want to simplify this : $$ \dfrac{-\sqrt{x}+\dfrac{\left( \sqrt{x}+1\right) ^{2}}{2\sqrt{x}}-1}{( \sqrt{x} + 1)^4} $$

to this :

$$ \dfrac{1-\sqrt{x}}{2\left( \sqrt{x}+1\right)^{3}\sqrt{x}} $$

I don't even know where to start. It would be nice if someone could show me each of the step required to arrive to the above expression while mentioning what rules are being used.

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    $\begingroup$ Start by factoring by $\sqrt x +1$ in the numerator. $\endgroup$
    – nejimban
    Oct 9, 2021 at 21:19
  • $\begingroup$ Start by multiplying numerator and denominator by$2\sqrt2$. $\endgroup$
    – Michael Hoppe
    Oct 10, 2021 at 8:56

1 Answer 1

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The numerator can easily be rewritten as follows: $\ - \sqrt{x} + \frac{(\sqrt{x} +1)^{2}}{2 \sqrt{x}} - 1 = - (\sqrt{x} + 1) + \frac{(\sqrt{x} +1)^{2}}{2 \sqrt{x}} = (\sqrt{x}+1)(-1 + \frac{\sqrt{x}+1}{2\sqrt{x}})$.

Now you can divide both numerator and denominator by $\sqrt{x} + 1$. If you then multiply numerator and denominator by $2 \sqrt{x}$, you will get the desired result.

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