Q:

Simple maths - rearranging terms

I have a formula:
$$\frac{\sqrt{\frac{2Ka}{h}}}{a}$$
How can it be arranged as:
$$\sqrt{\frac{2K}{ah}}$$
I only can do:
$$\frac{\sqrt{\frac{2Ka}{h}}}{a}$$
$$=\sqrt{\frac{2Ka}{h}}.\frac{1}{a}$$
$$=\frac{2Ka}{h}.\frac{1}{a^2}$$
$$=\frac{2K}{ah}$$
which is not correct, anyone help please?

A:

Write $a = \sqrt{a^2}$ and use the fact that $\sqrt{x/y} = \sqrt{x}/\sqrt{y}$ when $x, y > 0$.  (I am assuming $a, h, K$ are positive)  So $$\frac{\sqrt{\frac{2Ka}{h}}}{a} =\frac{\sqrt{\frac{2Ka}{h}}}{\sqrt{a^2}} = \sqrt{\frac{(\frac{2Ka}{h})}{a^2}} = \sqrt{\frac{2K}{ha}}   $$

A:

When $a$ is positive you can do $a=\sqrt{a^2}\;$ and you get 
$$\frac{\sqrt{\frac{2Ka}h}}a=\frac{\sqrt{\frac{2Ka}h}}{\sqrt{a^2}}=\sqrt{\frac{\frac{2Ka}h}{a^2}}=\sqrt{\frac{2Ka}{ha^2}}=\sqrt{\frac{2K}{ha}}$$

A:

There is a subtle distinction here that many students miss.
If I have an equation like $x = y$, then it is valid to square both sides and conclude that $x^2 = y^2$. Notice there are two sides of the equation to manipulate, so it makes sense to manipulate them simultaneously. The use of the equals sign here is to indicate that there are two different quantities, but they happen to have the same numerical value.
Your initial question is to simplify a single expression having no equals sign. Your use of the equals sign here is meant to indicate that you have multiple ways of writing your single initial quantity. There are not two different quantities in balance, so there is no sense in which you may "square both sides". What sides? There is only one quantity.
The point of all that is to say that you went awry at the second to last line. Squaring a single quantity changes that quantity.
If you want to bring the $\frac{1}{a}$ into the radical, try rewriting it as $\sqrt\frac{1}{a^2}$.

