How do you edit an expression on the right to accommodate the removal of a square root on the left.

Question

The $\sqrt{Z}$ is a multiplier for the constant on the right, how can I rewrite the constant to remove the square root from the multiplier and make the multiplier just $Z$? $\sqrt{Z}*(\frac{2\sqrt{\frac{\hbar}{SG}}}{\pi r^2})$

Thank you.

Answer 1

Changing from $\sqrt{Z}$ to Z on the left, you are multiplying by $\sqrt{Z}$. To correct for that you need to divide by $\sqrt{Z}$ on the right:$Z\left(\frac{2\sqrt{\frac{h}{SG}}}{\pi r^2\sqrt{Z}}\right)$.

If by "constant" you meant "not depending on Z" there is no way to keep that constant.

Answer 2

We can get rid of the $Z$-radical alone but not the corresponding radical for the expression as a whole.

Given $\quad \left(\dfrac{2\sqrt{\dfrac{\hbar}{SG}}}{\pi r^2}\right) = \dfrac{2\sqrt{h}}{\pi r^2\sqrt{SG}} = C \quad \text{The expression yields a constant value C.} $

$$\dfrac{2\sqrt{h}}{\pi r^2\sqrt{SG}}=C \implies \sqrt{Z}\dfrac{2\sqrt{h}}{\pi r^2\sqrt{SG}}=\sqrt{Z}C$$

$$\sqrt{Z}C\space = \space \sqrt{ZC^2} =\sqrt{Z\bigg(\dfrac{2\sqrt{h}}{\pi r^2\sqrt{SG}}\bigg)^2}$$