Question

The product (32), (32)^1/6 (32)^1/36 ... to ∞ is equal to
(a) 64
(b) 16
(c) 32
(d) 0

Answer 1

(a) 64

$$\begin{gathered} 32\times {32}^{\frac{1}{6}}\times {32}^{\frac{1}{36}}\times ...\infty \\ ={32}^{\left(1+\frac{1}{6}+\frac{1}{36}+...\infty \right)} \\ ={32}^{\left(\frac{1}{1-\frac{1}{6}}\right)}[\text{∵}\mathrm{it}\mathrm{is}\mathrm{a}\mathrm{G}.\mathrm{P}.] \\ ={32}^{\left(\frac{6}{5}\right)} \\ ={\left(2^5\right)}^{\left(\frac{6}{5}\right)} \\ =2^6 \\ =64 \end{gathered}$$