Why does $\dfrac{8}{\frac{8\sqrt{145}}{145}} = \sqrt{145}$?

Question

I can't seem to work out why this is true: $$\frac{8}{\dfrac{8\sqrt{145}}{145}} = \sqrt{145}$$

Could someone break it down for me?

Answer 1

Hint $\ \ \dfrac{8}{8 x}\, =\, \dfrac{1}x\,\ $ so $\,\ x = \dfrac{\sqrt{n}}n\ \Rightarrow\ \dfrac{1}x\, = \, \dfrac{n}{\sqrt{n}}\, =\, \dfrac{\sqrt{n}^2}{\sqrt{n}}\, =\, \sqrt{n}$

Answer 2

\begin{align} \frac{8}{\frac{8\sqrt{145}}{145}}&=\frac{8}{\frac{8}{\sqrt{145}}} & \text{because }\frac{\sqrt{145}}{145}=\frac 1{\sqrt{145}} \\ &=8\cdot\frac{\sqrt{145}}8 & \text{take reciprocal} \\ &=\sqrt{145} & \text{the $8$'s cancel} \end{align}

Answer 3

$\frac{8}{\frac{8\sqrt{145}}{145}}$ = $8\cdot \frac{145}{8\sqrt{145}}$.

Here, the $8$s cancel. So,

$\frac{145}{\sqrt{145}} = 145^1\cdot 145^{-\frac{1}{2}} = 145^{\frac{1}{2}} = \sqrt{145}$.

Answer 4

$8\left(\dfrac{145}{8\sqrt{145}}\right)=\dfrac{145}{\sqrt{145}}=\dfrac{145\sqrt{145}}{145}=\sqrt{145}$