Problem 139 - College Algebra

Question

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

Step-by-Step Solution

Answer

The given statement is false. Correct statement should be: \( 5^{2} \cdot 5^{-2} = 2^{5} \cdot 2^{-5} \). Both sides are equal to 1.

Step 1: Simplify both sides

We need to simplify both sides of the inequality to be able to compare them. Power laws allow us to do this. Recall that any base raised to the power of -a is equal to 1 divided by the base raised to the power of a, in this case: \( 5^{-2} = 1 / 5^{2} \) and \( 2^{-5} = 1 / 2^{5} \). Now we can apply these.

Step 2: Applying the power laws

Application of the power laws gives \( 5^{2} \cdot 5^{-2} = 5^{2} / 5^{2} = 1 \) and \( 2^{5} \cdot 2^{-5} = 2^{5} / 2^{5} = 1 \). So, the inequality is now \( 1 > 1 \) which is clearly incorrect.

Step 3: Correction of the false statement

Since we ended up with a false statement, the necessary change to produce a true statement would be to change the '>' into '=', because, as we already showed, both sides of the inequality indeed equal 1

Key Concepts

Understanding Inequalities

Inequalities allow us to compare different quantities. In this problem, we had an inequality that initially seemed to suggest one side was greater than the other. To deal with inequalities, you always aim to simplify both sides as much as possible. This can help you clearly see which side is bigger, or in some cases, find out that they are actually equal. When both sides of an inequality simplify to the same value, as it happened here where both became 1, the inequality action needs to be corrected. Instead of having a 'greater than' (>) symbol, you would replace it with an 'equals' sign (=). This correction is crucial because inequalities are about maintaining logical accuracy in mathematical statements. Simplifying helps you find out if an initial inequality holds true or which alterations can turn it into a true statement.

Simplification with Exponents

Simplification is the process of reducing an expression to its most basic form. It makes complex problems easier to solve or compare, like in this exercise. Here, we used exponent rules to simplify expressions involving exponential terms. By understanding that a negative exponent means a reciprocal, we can simplify expressions such as \( 5^{-2} = \frac{1}{5^{2}} \). This understanding is crucial because once we rewrite expressions with negative exponents as fractions, it becomes easier to multiply or divide them just like regular numbers. Applying these rules led us to see that \( 5^{2} \cdot 5^{-2} = \frac{5^{2}}{5^{2}} \), which simplifies to 1. Similarly, the other side also simplified to 1. This way, we could easily see that the two sides should be equal, not unequal.

Power Laws and Their Application

Power laws are fundamental rules that govern operations involving powers or exponents. These rules help simplify expressions such as those in inequalities. One of the key power laws is the product of powers property: \( a^{m} \cdot a^{-m} = a^{m-m} = a^{0} = 1 \). This law was essential in solving the given problem. By using this property, we concluded that both sides of the inequality simplify to 1. Another useful power law is the exponential property of zero: any non-zero number raised to the power of zero equals one, \( a^{0} = 1 \). Understanding and correctly applying these laws helps significantly when dealing with more complex expressions containing exponents. These rules are like super tools helping reduce and simplify big exponential problems to simple calculations or results, making it obvious if any adjustments in inequalities are needed.