<p>\( x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} \cdot x^{\frac{1}{8}} \cdot x^{\frac{1}{16}} \cdots \) to \( \infty \) is equal to</p>
Step-by-Step Solution
Key Concept: The exponents form a geometric series: 1/2 + 1/4 + 1/8 + 1/16 + ... = 1/(1-1/2) = 1. Therefore x^(1/2 + 1/4 + 1/8 + ...) = x^1 = x.
<p><strong>Step 1:</strong> Use the law of exponents: x^a · x^b · x^c · ... = x^(a+b+c+...)</p><p><strong>Step 2:</strong> Identify the exponents: 1/2, 1/4, 1/8, 1/16, ... form a geometric series with first term a = 1/2 and common ratio r = 1/2</p><p><strong>Step 3:</strong> Apply the infinite geometric series formula: Sum = a/(1-r) = (1/2)/(1-1/2) = (1/2)/(1/2) = 1</p><p><strong>Step 4:</strong> Therefore: x^(1/2 + 1/4 + 1/8 + ...) = x^1 = x</p><p>∴ Answer: <strong>x</strong></p>
Correct Answer: C