# fractions with powers

• Nov 28th 2011, 11:34 PM
Jskid
fractions with powers
How does $\displaystyle 1-\frac{1}{3^{k-1}}+\frac{2}{3^k}=1- \frac{1}{3^k}$?
I tried putting it over a common denominator and got $\displaystyle 1-\frac{3^k+6^{k-1}}{3^{k-1} \cdot 3^k}$
• Nov 29th 2011, 12:27 AM
princeps
Re: fractions with powers
Quote:

Originally Posted by Jskid
How does $\displaystyle 1-\frac{1}{3^{k-1}}+\frac{2}{3^k}=1- \frac{1}{3^k}$?
I tried putting it over a common denominator and got $\displaystyle 1-\frac{3^k+6^{k-1}}{3^{k-1} \cdot 3^k}$

$\displaystyle 1-\frac{1}{3^{k-1}}+\frac{2}{3^k}=1-\frac{1}{3^{k-1}}+\frac{2}{3\cdot 3^{k-1}}=1-\left(\frac{1}{3^{k-1}}-\frac{2}{3\cdot 3^{k-1}}\right)=$

$\displaystyle =1-\left(\frac{3-2}{3\cdot 3^{k-1}}\right)=1-\frac{1}{3^k}$
• Nov 29th 2011, 04:17 AM
Soroban
Re: fractions with powers
Hello, Jskid!

A slightly different approach . . .

Quote:

$\displaystyle \text{How does }\,1-\frac{1}{3^{k-1}}+\frac{2}{3^k}\:=\:1- \frac{1}{3^k}\:?$

Consider the two fractions: .$\displaystyle - \frac{1}{3^{k-1}} + \frac{2}{3^k}$

Multiply the first fraction by $\displaystyle \tfrac{3}{3}\!:\;\;-\frac{1}{3^{k-1}}\cdot\frac{3}{3}\;+\;\frac{2}{3^k}$

Then we have: .$\displaystyle -\frac{3}{3^k} + \frac{2}{3^k} \;=\;\frac{-3 + 2}{3^k} \;=\;-\frac{1}{3^k}$