What is the following value when expressed as a common fraction: $$\frac{1}{3^{1}}+\frac{1}{3^{2}}+\frac{1}{3^{3}}+\frac{1}{3^{4}}+\frac{1}{3^{5}}+\frac{1}{3^{6}}?$$
To find a common denominator, I can use the least common multiple of the powers of 3. The least common multiple of 3, 9, 27, 81, 243, and 729 is 729, since it is the highest power of 3. So I can rewrite each fraction with a denominator of 729 by multiplying the numerator and denominator by the appropriate power of 3. For example, $\frac{1}{3^{1}}=\frac{3^{5}}{3^{6}}=\frac{243}{729}$. Similarly, $\frac{1}{3^{2}}=\frac{3^{4}}{3^{6}}=\frac{81}{729}$, and so on. So the sum becomes: $$\frac{243}{729}+\frac{81}{729}+\frac{27}{729}+\frac{9}{729}+\frac{3}{729}+\frac{1}{729}.$$ The sum of the numerators is: $243+81+27+9+3+1=364$. So the final fraction is: $\frac{364}{729}$. 