Question

Evaluate \(25^{-\frac{1}{2}}\) A. \(-\frac{25}{2}\) B. \(\frac{1}{625}\) C. \(\frac{1}{5}\) D. -625

Short answer

Option C \(\frac{1}{5}\) is the correct answer.

Step-by-step solution

Understand the exponent

The expression to evaluate is \(25^{-\frac{1}{2}}\). The exponent \(-\frac{1}{2}\) indicates a reciprocal and a square root. Specifically, \(x^{-\frac{1}{2}} = \frac{1}{\sqrt{x}}\).

Apply the exponent rule

Using the rule from Step 1, rewrite the expression \(25^{-\frac{1}{2}}\) as \(\frac{1}{25^{\frac{1}{2}}}\).

Simplify the square root

Recognize that \(25^{\frac{1}{2}}\) is the same as the square root of 25. So, \(25^{\frac{1}{2}} = \sqrt{25}\).

Calculate the square root

The square root of 25 is 5, so \(\sqrt{25} = 5\).

Write the final expression

The expression now is \(\frac{1}{5}\).

Verify the answer from options

Compare the calculated answer \(\frac{1}{5}\) with the given options. Option C \(\frac{1}{5}\) is the correct answer.

Key concepts

Understanding Reciprocals

The concept of a reciprocal is quite fundamental in math. The reciprocal of a number is essentially one divided by that number. For example, the reciprocal of 5 is \(\frac{1}{5}\). This concept becomes very important when dealing with negative exponents. A negative exponent like \(-\frac{1}{2}\) tells you to take the reciprocal of the base. In our exercise, evaluating \(25^{-\frac{1}{2}}\) can be viewed as first recognizing it's a reciprocal: \(25^{-\frac{1}{2}} = \frac{1}{25^{\frac{1}{2}}}\). This action of flipping the base to the denominator is what the negative sign indicates.

Using Square Roots

A square root is another vital concept. Taking the square root of a number means finding a value that, when multiplied by itself, gives you the original number. For example, \(\text{the square root of 25 is 5 because } 5 \times 5 = 25\). In the exercise, understanding that \(\frac{1}{2}\) in the exponent means a square root is key. When we see \({25^{\frac{1}{2}}}\), it is just another way to write \(\text{the square root of 25}\). Thus, \({25^{\frac{1}{2}}=\text{5}}\).

Simplifying Exponents

Simplifying exponents can seem daunting at first, but it's all about following some key rules. Rules around exponents help us break down complex expressions into simpler ones we can evaluate easily. In the step-by-step solution, we used the fact that \(25^{-\frac{1}{2}}\) simplifies to \(\frac{1}{25^{\frac{1}{2}}}\). This rule basically says a negative exponent creates a reciprocal of the base raised to the opposite (positive) power. After this, we applied the understanding of square roots. Following these rules and breaking down the steps methodically leads us to the conclusion that \(\frac{1}{25^{\frac{1}{2}}} = \frac{1}{5}\).