Question

If $\frac{1}{{\left(243\right)}^x}={\left(729\right)}^y=3^3$, then 5x + 6y = __________.

Answer 1

$$\begin{gathered} \mathrm{Given}:\frac{1}{{\left(243\right)}^x}={\left(729\right)}^y=3^3 \\ \frac{1}{{\left(243\right)}^x}=3^3 \\ \Rightarrow \frac{1}{{\left(3^5\right)}^x}=3^3 \\ \Rightarrow \frac{1}{3^{5x}}=3^3 \\ \Rightarrow 3^{-5x}=3^3 \\ \Rightarrow -5x=3 \\ \Rightarrow 5x=-3...\left(1\right) \\ {\left(729\right)}^y=3^3 \\ \Rightarrow {\left(3^6\right)}^y=3^3 \\ \Rightarrow 3^{6y}=3^3 \\ \Rightarrow 6y=3...\left(2\right) \\ \mathrm{Adding}\left(1\right)\mathrm{and}\left(2\right),\mathrm{we}\mathrm{get} \\ 5x+6y=-3+3 \\ =0 \end{gathered}$$

Hence, 5x + 6y = 0.