Question

The value of the product (32) × (32)^1/6 × (32)^1/36 × _________ to ∞, is _________.

Answer 1

$$\begin{gathered} \left(32\right)\times {\left(32\right)}^{\frac{1}{6}}\times {\left(32\right)}^{\frac{1}{36}}\times .........\infty \\ ={\left(32\right)}^{\left(1+\frac{1}{6}+\frac{1}{36}+.....\right)} \\ \mathrm{Since}1+\frac{1}{6}+\frac{1}{6^2}+.......\mathrm{from}\mathrm{an}\mathrm{g}.\mathrm{p} \end{gathered}$$
$$\begin{gathered} ={32}^{\left(\frac{1}{1-{\displaystyle \frac{1}{6}}}\right)} \\ ={\left(32\right)}^{\left(\frac{6}{5}\right)} \\ ={\left(2^5\right)}^{\frac{6}{5}} \\ =2^{5\times \frac{6}{5}} \\ =2^6=64 \\ \mathrm{i}.\mathrm{e}\left(32\right)\times {\left(32\right)}^{\frac{1}{6}}\times {\left(32\right)}^{\frac{1}{36}}.........\infty \end{gathered}$$