Question

which rational number can be multiplied to 1/ (3^3 * 5^4)^1/7 so that the denominator becomes a rational number?

Answer 1

$$\begin{gathered} \mathrm{Dear}\mathrm{Student}, \\ \frac{1}{{\left(3^3\times 5^4\right)}^{{\displaystyle \frac{1}{7}}}}=\frac{1}{\left(3^{{\displaystyle \frac{3}{7}}}\times 5^{{\displaystyle \frac{4}{7}}}\right)} \\ \mathrm{Now}\mathrm{to}\mathrm{make}\mathrm{denominator}\mathrm{rational}\mathrm{we}\mathrm{should}\mathrm{make}\mathrm{power}\mathrm{an}\mathrm{integer} \\ \mathrm{so}\mathrm{we}\mathrm{should}\mathrm{multiply}\mathrm{it}\mathrm{by}{3}^{\frac{4}{7}}\times 5^{\frac{3}{7}}. \\ =\frac{1}{\left(3^{{\displaystyle \frac{3}{7}}}\times 5^{{\displaystyle \frac{4}{7}}}\right)}\times \frac{3^{\frac{4}{7}}\times 5^{\frac{3}{7}}}{3^{\frac{4}{7}}\times 5^{\frac{3}{7}}}=\frac{3^{\frac{4}{7}}\times 5^{\frac{3}{7}}}{3^{\frac{4}{7}+{\displaystyle \frac{3}{7}}}\times 5^{\frac{3}{7}+{\displaystyle \frac{4}{7}}}} \\ =\frac{3^{\frac{4}{7}}\times 5^{\frac{3}{7}}}{3^1\times 5^1}=\frac{3^{\frac{4}{7}}\times 5^{\frac{3}{7}}}{15} \\ \mathrm{Regards} \end{gathered}$$