How Do You Simplify Complex Fractions with Nested Radicals?

[russdot]

Homework Statement

$$\frac{1 + \sqrt{\frac{1}{1 + \left(\frac{s}{u}\right)^{2}}}}{1 - \sqrt{\frac{1}{1 + \left(\frac{s}{u}\right)^{2}}}}$$

Should equal $$\left(\sqrt{1 + \left(\frac{u}{s}\right)^2} + \left(\frac{u}{s}\right)\right)^{2}$$

Homework Equations

above

The Attempt at a Solution

I have tried numerous ways of trying to simplify the initial equation to equal the second, but cannot seem to get rid of the multi-term denominator.

What is a good formula/method for reducing a fraction such as this?
Thanks

 

[Avodyne]

When you have a denominator of the form $1-\sqrt{A}$, multiply numerator and denominator by $1+\sqrt{A}$.

 

[Architeuthis Dux]

for your question. Simplifying complex fractions can be a challenging task, but there are some general strategies that can be used to simplify them. One approach is to try to find common factors in the numerator and denominator, and then cancel them out. In this particular case, we can start by simplifying the square roots in the numerator and denominator:

\frac{1 + \sqrt{\frac{1}{1 + \left(\frac{s}{u}\right)^{2}}}}{1 - \sqrt{\frac{1}{1 + \left(\frac{s}{u}\right)^{2}}}} = \frac{1 + \frac{1}{\sqrt{1 + \left(\frac{s}{u}\right)^{2}}}}{1 - \frac{1}{\sqrt{1 + \left(\frac{s}{u}\right)^{2}}}}

Next, we can use the fact that \sqrt{\frac{1}{x}} = \frac{1}{\sqrt{x}} to rewrite the fractions in a more manageable form:

\frac{1 + \frac{1}{\sqrt{1 + \left(\frac{s}{u}\right)^{2}}}}{1 - \frac{1}{\sqrt{1 + \left(\frac{s}{u}\right)^{2}}}} = \frac{\frac{\sqrt{1 + \left(\frac{s}{u}\right)^{2}}}{\sqrt{1 + \left(\frac{s}{u}\right)^{2}}} + \frac{1}{\sqrt{1 + \left(\frac{s}{u}\right)^{2}}}}{\frac{\sqrt{1 + \left(\frac{s}{u}\right)^{2}}}{\sqrt{1 + \left(\frac{s}{u}\right)^{2}}} - \frac{1}{\sqrt{1 + \left(\frac{s}{u}\right)^{2}}}}

Now, we can combine the fractions in the numerator and denominator by finding a common denominator, which is \sqrt{1 + \left(\frac{s}{u}\right)^{2}}:

\frac{\frac{\sqrt{1 + \left(\frac{s}{u}\right)^{2}}}{\sqrt{1 + \left(\frac{s}{u}\right)^{2}}} + \frac{1}{\sqrt{1 + \left(\frac{s