We are asked to simplify the following expression:

$\left(\sqrt{x}\right) \left(\frac{1}{2\sqrt{x}}\right)$

Step-by-Step Simplification:

1. Combine the terms: $\sqrt{x} \times \frac{1}{2\sqrt{x}} = \frac{\sqrt{x} \cdot 1}{2\sqrt{x}}$ This simplifies to: $\frac{\sqrt{x}}{2\sqrt{x}}$

2. Simplify the fraction: Since $\sqrt{x}$ is in both the numerator and denominator, they cancel out: $\frac{\sqrt{x}}{2\sqrt{x}} = \frac{1}{2}$

Thus, the simplified form of the expression is:

$\boxed{\frac{1}{2}}$

Would you like me to go deeper into any of these steps?

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Here are 5 related questions:

1. How do you simplify other algebraic fractions involving square roots?
2. What are some properties of square roots that help in simplification?
3. How do you multiply and divide terms involving radicals?
4. How would this expression change if it were $\sqrt{x^2}$ instead of $\sqrt{x}$?
5. How do we rationalize fractions with radicals in the denominator?

Tip: Always check for common terms in the numerator and denominator when simplifying fractions—they might cancel!