Problem 137 - College Algebra Essentials

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$5^{-2}>2^{-5}$$

Step-by-Step Solution

Answer

The statement \(5^{-2}>2^{-5}\) is true.

Step 1: Understanding the exponential notation

When dealing with negative exponents, keep in mind the rule that \(a^{-n} = \frac{1}{a^n}\). Here, \(a\) is the base and \(n\) is the negative exponent. Applying this to the given statement, \(5^{-2}\) will become \(\frac{1}{5^2}\) and \(2^{-5}\) will turn into \(\frac{1}{2^5}\)

Step 2: Calculate values

Next, calculate the values of these expressions. For \(\frac{1}{5^2}\), it can be evaluated as \(\frac{1}{25} = 0.04\). As for \(\frac{1}{2^5}\), it turns out to be \(\frac{1}{32} = 0.03125\)

Step 3: Compare and confirm

With the evaluations of \(5^{-2}\) and \(2^{-5}\), check if the inequality \(5^{-2}>2^{-5}\) holds true, i.e., if \(0.04 > 0.03125\). Since 0.04 is indeed greater than 0.03125, the statement is true.