Solve (frac{9^{n}+27^n}{3^n+9^n})^frac{1{2}}

$$( \frac { 9 ^ { n } + 27 ^ { n } } { 3 ^ { n } + 9 ^ { n } } ) ^ { \frac { 1 } { 2 } }$$

Evaluate

$$3^{\frac{n}{2}}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{9^{n}+27^{n}}{3^{n}+9^{n}}.

$$\left(\frac{\left(3^{n}+1\right)\times \left(3^{n}\right)^{2}}{3^{n}\left(3^{n}+1\right)}\right)^{\frac{1}{2}}$$

Cancel out 3^{n}+1 in both numerator and denominator.

$$\left(\frac{\left(3^{n}\right)^{2}}{3^{n}}\right)^{\frac{1}{2}}$$

To raise \frac{\left(3^{n}\right)^{2}}{3^{n}} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{\left(\left(3^{n}\right)^{2}\right)^{\frac{1}{2}}}{\left(3^{n}\right)^{\frac{1}{2}}}$$

To raise a power to another power, multiply the exponents. Multiply 2 and \frac{1}{2} to get 1.

$$\frac{\left(3^{n}\right)^{1}}{\left(3^{n}\right)^{\frac{1}{2}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract \frac{1}{2} from 1 to get \frac{1}{2}.

$$\left(3^{n}\right)^{\frac{1}{2}}$$

Differentiate w.r.t. n

$$\frac{\ln(3)\times 3^{\frac{n}{2}}}{2}$$