Question

Simplify each expression. $$-25^{-1 / 2}$$

Short answer

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

Step-by-step solution

Understand the Negative Exponent

A negative exponent indicates the reciprocal of the base raised to the positive version of the exponent. Therefore, for \(-25^{-1/2}\), we can rewrite it as \(\frac{1}{25^{1/2}}\).

Simplify the Exponent

Next, we simplify the expression \(\frac{1}{25^{1/2}}\). The \(1/2\) exponent means taking the square root. Hence, \(25^{1/2} = \sqrt{25} = 5\).

Complete the Simplification

Substitute back the simplified value: \(\frac{1}{5}\).

Key concepts

Negative Exponents

When you encounter a negative exponent, it might seem confusing at first. But don't worry; it's simpler than it looks! A negative exponent tells you to take the reciprocal (or inverse) of the base number and change the exponent to positive.
For example, \(a^{-n} = \frac{1}{a^n}\).
In the exercise, we had \(-25^{-1/2}\). This can be rewritten as \(- \frac{1}{25^{1/2}}\) because the negative exponent means we take the reciprocal of \25\ and flip the exponent sign to positive.
This basic rule can help you simplify many other expressions with negative exponents.

Reciprocal

The reciprocal of a number is simply 1 divided by that number. So, if you have the number \ a \, its reciprocal is \ \frac{1}{a} \.
Reciprocals are very useful in simplifying negative exponents. For instance, consider \(a^{-n}\). You would find the reciprocal of \a\ and then raise it to the power of \, as in \(a^{-n} = \frac{1}{a^n}\).
In our example, for the \ -25^{-1/2} \, after taking the reciprocal of 25, it became \ - \frac{1}{25^{1/2}} \. This means we take the reciprocal first and then simplify further.

Square Root

A square root is a value that, when multiplied by itself, gives the original number. The square root symbol is \sqrt{}\.
For example, the square root of 25 is 5 because \(5 \times 5 = 25\). Mathematically, we write \sqrt{25} = 5\.
In our exercise, we used the square root when simplifying \25^{1/2}\ because the exponent \(1/2\) indicates the square root operation. So, \25^{1/2} = \sqrt{25} = 5\.
Finally, we substituted this back into our expression to get \ - \frac{1}{5} \.