# Simplification of an equation based on exponent rules

#### ATroelstein

##### New member
I have the following equation that I'm trying to simplify:

$$\frac{5 + \sqrt{5}}{2\sqrt{5}}*(\frac{1 + \sqrt{5}}{2})^{x}$$

From looking at it, it seems like it could be simplified so that the right-hand side of the multiplication would be:

$$(\frac{1 + \sqrt{5}}{2})^{x+1}$$

I started to pull the left-hand side apart to get a match to the right-hand side and ended up with:

$$(\frac{4}{2\sqrt{5}} + \frac{1}{\sqrt{5}} * \frac{1+\sqrt{5}}{2}) * (\frac{1 + \sqrt{5}}{2})^{x}$$

From here, I'm starting to wonder if my initial observation was flawed. Is there a way to simplify this is terms of:

$$(\frac{1 + \sqrt{5}}{2})^{x+1}$$

Thanks.

#### mathbalarka

##### Well-known member
MHB Math Helper
$\displaystyle \frac{5 + \sqrt{5}}{2 \sqrt{5}}$ = $\displaystyle \frac{1 + \sqrt{5}}{2}$ by multiplying numerator and the denominator by $$\displaystyle \sqrt{5}$$

Hence, the multiplication of it with $$\displaystyle \left ( \frac{1 + \sqrt{5}}{2} \right )^x$$ is equal to $$\displaystyle \left ( \frac{1 + \sqrt{5}}{2} \right )^{x+1}$$

#### MarkFL

$$\displaystyle \frac{5+\sqrt{5}}{2\sqrt{5}}$$