The value of 5^{1/2} cdot 5^{1/4} cdot 5^{1/8 | vedclass

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(D) Let the given expression be $X = 5^{1/2} \cdot 5^{1/4} \cdot 5^{1/8} \cdots \infty$. Since the bases are the same,we add the exponents: $X = 5^{(1

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$5$

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(D) Let the given expression be $X = 5^{1/2} \cdot 5^{1/4} \cdot 5^{1/8} \cdots \infty$. Since the bases are the same,we add the exponents: $X = 5^{(1/2 + 1/4 + 1/8 + \cdots \infty)}$. The exponent is an infinite geometric series with first term $a = 1/2$ and common ratio $r = 1/2$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$. Substituting the values,$S = \frac{1/2}{1 - 1/2} = \frac{1/2}{1/2} = 1$. Therefore,$X = 5^1 = 5$.

The value of $5^{1/2} \cdot 5^{1/4} \cdot 5^{1/8} \cdots \infty$ is .......

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