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

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Question

(B) The given expression is a product of powers with the same base $32$.Using the property $a^m \cdot a^n = a^{m+n}$, we can write the product as:$(32

Vedclass · Accepted Answer

$64$

Vedclass · Answer

(B) The given expression is a product of powers with the same base $32$.Using the property $a^m \cdot a^n = a^{m+n}$, we can write the product as:$(32)^{1 + 1/6 + 1/36 + \ldots + \infty} = (32)^x$where $x = 1 + 1/6 + 1/36 + \ldots + \infty$.This is an infinite geometric series with the first term $a = 1$ and common ratio $r = 1/6$.The sum of an infinite geometric series is given by $S = \frac{a}{1-r}$.Substituting the values, $x = \frac{1}{1 - 1/6} = \frac{1}{5/6} = 6/5$.Therefore, the product is $(32)^{6/5}$.Since $32 = 2^5$, we have $(2^5)^{6/5} = 2^{5 \cdot (6/5)} = 2^6$.$2^6 = 64$.

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

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