# Simplifying an expression with exponents

I'm currently working on a physics problem and I have worked out the problem to the reach the following expression

$$\dfrac{2\epsilon(e^{-\epsilon\beta}+e^{-2\epsilon \beta})}{1+2e^{-\epsilon \beta}+ e^{-2 \epsilon \beta}}$$

Is there a way that I can turn the expression that I have into this

$$\dfrac{2\epsilon}{1+e^{\epsilon \beta}}$$

This is the answer in the book and I don't have a clue how it got there to be honest.

EDIT: Thanks very much guys!!

• Mathematica simplifies the expression immediately. – David G. Stork Oct 5 '17 at 21:14

$$\frac{2\epsilon(e^{-\epsilon\beta}+e^{-2\epsilon \beta})}{1+2e^{-\epsilon \beta}+ e^{-2 \epsilon \beta}}$$

$$= \left(\frac{2\epsilon(e^{-\epsilon\beta}+e^{-2\epsilon \beta})}{1+2e^{-\epsilon \beta}+ e^{-2 \epsilon \beta}} \right) \frac{e^{2 \epsilon \beta}}{e^{2 \epsilon \beta}}$$

$$= \frac{2\epsilon(e^{\epsilon\beta}+1)}{e^{2 \epsilon \beta}+2e^{\epsilon \beta}+ 1}$$

$$= \frac{2\epsilon(e^{\epsilon\beta}+1)}{(e^{\epsilon \beta}+1)^2}$$

$$= \frac{2\epsilon}{(1+e^{\epsilon \beta})}$$

$$\dfrac{2\epsilon(e^{-\epsilon\beta}+e^{-2\epsilon \beta})}{1+2e^{-\epsilon \beta}+ e^{-2 \epsilon \beta}} =\dfrac{2\epsilon(e^{-\epsilon\beta}+e^{-2\epsilon \beta})}{1+2e^{-\epsilon \beta}+ e^{-2 \epsilon \beta}}\times \frac{ e^{2 \epsilon \beta}}{e^{2 \epsilon \beta}} =\dfrac{2\epsilon(e^{\epsilon\beta}+1)}{1+2e^{\epsilon \beta}+ e^{2 \epsilon \beta}} =\dfrac{2\epsilon(e^{\epsilon\beta}+1)}{(1+e^{\epsilon \beta})^2} =\dfrac{2\epsilon}{1+e^{\epsilon \beta}}$$

HINT: Multiply numerator and denominator both by $e^{2\epsilon \beta}$ and try to simplify.

Hope this helps you.