How to simplify fractions involving surds? $\frac{(\sqrt{θ^2+1})(2θ)-(θ^2-1)(\frac{θ}{\sqrt{θ^2+1}})}{θ^2+1}$

Question

I came across this example of the Quotient Rule and can't understand how it was simplified from step $2$ to $3$. How were the roots turned into the power of $3/2$?

Find $\frac{du}{d\theta}$ when $u = \frac{\theta^2 −1} {\sqrt{\theta^2 + 1}}$

1) $$ \frac{du}{d\theta} = \frac{w\frac{dv}{d\theta} − v \frac{dw}{d\theta}} {w^2} = $$

2) $$ \frac{(\sqrt{\theta^2+1})(2\theta)-(\theta^2-1)(\frac{\theta}{\sqrt{\theta^2+1}})}{\theta^2+1}= $$

3) $$ \frac{(\theta^2 + 1)(2\theta)−(\theta^2 −1)(\theta)}{(\theta^2 + 1)^{3/2}}= $$

4) $$ \frac{2\theta^3 + 2\theta−\theta^3 + \theta}{(\theta^2 + 1)^{3/2}}= $$

5) $$ \frac{\theta(\theta^2 + 3)}{(\theta^2 + 1)^{3/2}} $$

Answer 1

From 2 to 3, multiply top and bottom by $\sqrt{\theta^2+1}$, then use $x\sqrt{x}=x^{3/2}$ with $x=\theta^2+1$.