2n4-find-times-frac-integer-n5

Question 73

Find the value of $n$ , where $n$ is an integer and $2^{n - 5} \times 6^{2 n - 4} = \frac{1}{12^{4} \times 2} .$

We have $2^{n - 5} \times 6^{2 n - 4} = \frac{1}{12^{4} \times 2} \Rightarrow \frac{2^{n}}{2^{5}} \times \frac{6^{2 n}}{6^{4}} = \frac{1}{12^{4} \times 2} \Rightarrow \frac{2^{n} \times 6^{2 n}}{2^{5} \times 6^{4}} = \frac{1}{(2 \times 6)^{4} \times 2} \Rightarrow 2^{n} \times (6^{2})^{n} = \frac{2^{5} \times 6^{4}}{2^{4} \times 6^{4} \times 2} \Rightarrow 2^{n} \times 36^{n} = \frac{2^{5} \times 6^{4}}{2^{5} \times 6^{4}} \Rightarrow 2^{n} \times 36^{n} = 1 \Rightarrow (2 \times 36)^{n} = 1 \Rightarrow (72)^{n} = (72)^{0} ∴ n = 0$