Verified Answer :

  • Use the power of a quotient rule \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) to expand the expression.
  • Compute the square of the numerator: \(1^2 = 1\).
  • Compute the square of the denominator: \(2^2 = 4\).
  • Match the result to the correct option: \(\boxed{\frac{1}{4}}\).

Explanation

  1. Understanding the Expression

The objective is to evaluate the expression \(\left(\frac{1}{2}\right)^2\) and select the correct answer from the given options. The expression consists of a fraction \(\frac{1}{2}\) raised to the power of 2. Applying the exponent rules, specifically the rule for the power of a quotient, \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\), we can break down the exponentiation of the numerator and the denominator separately.

  1. Calculating the Power

Applying the rule \(\left(\frac{1}{2}\right)^2 = \frac{1^2}{2^2}\).

  • The numerator is \(1^2 = 1 \times 1 = 1\).
  • The denominator is \(2^2 = 2 \times 2 = 4\).

Thus, the value of the expression is \(\frac{1}{4}\).

  1. Concluding the Calculation

Comparing the calculated value \(\frac{1}{4}\) to the provided options:

A\(\frac{2}{25}\)
B\(\frac{1}{4}\)
C\(1\)
D\(\frac{1}{8}\)

The result matches option (b). Therefore, the correct choice is b. The final value is \(\boxed{\frac{1}{4}}\).

Examples

If you have a square with a side length of \(\frac{1}{2}\) meter, its area is calculated by squaring the side length, i.e., \(\left(\frac{1}{2} \text{ m}\right)^2 = \frac{1}{4} \text{ m}^2\). This demonstrates how exponentiation with fractions applies to geometric measurements like area.

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