The product (32)(32)^{1/6}(32)^{1/36} dots in | vedclass

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Question

(C) Let the given expression be $P = (32)(32)^{1/6}(32)^{1/36} \dots \infty$.Using the property of exponents $a^m 	imes a^n = a^{m+n}$,we have:$P = (

Vedclass · Accepted Answer

$64$

Vedclass · Answer

(C) Let the given expression be $P = (32)(32)^{1/6}(32)^{1/36} \dots \infty$.Using the property of exponents $a^m 	imes a^n = a^{m+n}$,we have:$P = (32)^{1 + \frac{1}{6} + \frac{1}{36} + \dots \infty}$.The exponent is an infinite geometric series with the first term $a = 1$ and common ratio $r = \frac{1}{6}$.The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$.$S = \frac{1}{1 - 1/6} = \frac{1}{5/6} = \frac{6}{5}$.Therefore,$P = (32)^{6/5}$.Since $32 = 2^5$,we have $P = (2^5)^{6/5} = 2^{5 	imes (6/5)} = 2^6$.$P = 64$.

The product $(32)(32)^{1/6}(32)^{1/36} \dots \infty$ is

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