Problem 139 - 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. $$5^{2} \cdot 5^{-2}>2^{5} \cdot 2^{-5}$$

Step-by-Step Solution

Answer

The statement given is false. To make it true, the inequality sign can be replaced by an equality sign, since both the expressions on either side of the inequality are equal.

Step 1: Simplify the Expressions

The first step requires understanding the rules of exponents. Any number raised to the power of a negative exponent can be converted into a fraction with the base of the exponent as the denominator, raised to the power of the absolute value of that exponent. Therefore, \(5^{-2}\) becomes \(\frac{1}{5^2}\) and \(2^{-5}\) becomes \(\frac{1}{2^5}\). The inequality now turns into \(5^2 \cdot \frac{1}{5^2} > 2^5 \cdot \frac{1}{2^5}\).

Step 2: Further Simplification and Comparison

On both sides of the inequality, there are multiplication operations between numbers with the same base but different exponents. According to another rule of exponents, these can be simplified into 1. Therefore, the inequality further simplifies into \(1 > 1\).

Step 3: Check the Truth Value

Comparing both sides of the inequality, it becomes evident that 1 is not greater than 1, so the original statement is false.