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Simplification of an equation based on exponent rules

ATroelstein

New member
Jun 30, 2012
15
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
Mar 22, 2013
573
$ \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

Administrator
Staff member
Feb 24, 2012
13,775
First, I want to say that what you have is an expression, not an equation. An equation gives the equality between two expressions, and is identified by the use of the "=" sign. I have seen many student confuse the two terms. :D

This may seem like a trivial point to make, but in order to be able to communicate effectively in any subject, the correct use of terminology is crucial.

Your expression consists ot two factors. Look at the factor on the left:

\(\displaystyle \frac{5+\sqrt{5}}{2\sqrt{5}}\)

Can you factor the numerator, so that you may cancel with a factor in the denominator?

I now see I have been pipped at the post, but do you see how mathbalarka simplified the factor on the left?