Problem 128 - Algebra A Combined Function

Question

Determine whether each statement is true or false. a. \(5^{-1}<5^{-2}\) b. \(\left(\frac{1}{5}\right)^{-1}<\left(\frac{1}{5}\right)^{-2}\) c. \(a^{-1}

Step-by-Step Solution

Answer

a. False b. True c. False

Step 1: Understanding Negative Exponents

Negative exponents indicate the reciprocal of a number raised to a positive exponent. For example, if you have a number with a negative exponent, such as \(x^{-n}\), it is equivalent to \(\frac{1}{x^n}\). Understanding this will help you evaluate which reciprocal is larger.

Step 2: Evaluating Statement a

Convert the negative exponents to positive by taking their reciprocals: \(5^{-1} = \frac{1}{5}\)\(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\)Compare \(\frac{1}{5}\) and \(\frac{1}{25}\). Since \(\frac{1}{5}\) is greater than \(\frac{1}{25}\), the statement \(5^{-1} < 5^{-2}\) is false.

Step 3: Evaluating Statement b

Convert into positive exponent reciprocals:\(\left(\frac{1}{5}\right)^{-1} = 5\)\(\left(\frac{1}{5}\right)^{-2} = 5^2 = 25\)Compare these results: Since \(5 < 25\), the statement \(\left(\frac{1}{5}\right)^{-1} < \left(\frac{1}{5}\right)^{-2}\) is true.

Step 4: Evaluating Statement c

Evaluate for nonzero \(a\):Rewrite using reciprocal logic:\(a^{-1} = \frac{1}{a}\)\(a^{-2} = \frac{1}{a^2}\)For all nonzero \(a\), \(\frac{1}{a}\) is greater than \(\frac{1}{a^2}\), so \(a^{-1} < a^{-2}\) is false.

Key Concepts

Reciprocal

The term "reciprocal" might sound a bit intimidating at first, but it's quite simple once you get the hang of it. When we talk about the reciprocal of a number, we mean the inverse of that number. In simpler terms, if you start with a number like 5, the reciprocal is found by dividing 1 by that number, giving you \( \frac{1}{5} \). This means multiplication of a number by its reciprocal results in 1.

This concept is quite handy when dealing with negative exponents. A negative exponent, like \( x^{-n} \), indicates the reciprocal, or the inverse, of a positive power. So, \( x^{-n} = \frac{1}{x^n} \).

This understanding is critical when working with expressions involving negative exponents, as it helps to simplify and solve equations or inequalities accurately.

Exponent Rules

Exponent rules, particularly those for negative exponents, simplify many algebraic expressions and their operations. The key idea is that negative exponents indicate a reciprocal. For instance, \( a^{-n} \) means \( \frac{1}{a^n} \). This flips the base to the opposite, or reciprocal side of the fraction.

Here are a few important rules to keep in mind:

• For any nonzero number \( a \), \( a^0 = 1 \).
• \( a^{-1} = \frac{1}{a} \).
• If you have \( \left( \frac{1}{a} \right)^{-n} = a^n \).

Understanding these rules helps you navigate equations and expressions both easily and correctly. When you encounter negative exponents, think "reciprocal"; it’s your first step towards simplifying complex calculations.

Comparison of Fractions

To compare fractions, especially those derived from negative exponents, start by translating the expressions into their reciprocal forms. For example, when looking at expressions like \( 5^{-1} = \frac{1}{5} \) and \( 5^{-2} = \frac{1}{25} \), it becomes easier to understand which is bigger or smaller by considering the value of the denominators.

When comparing fractions:

• The larger the denominator, the smaller the fraction if the numerators are equal. For instance, \( \frac{1}{25} \) is less than \( \frac{1}{5} \) because 25 is larger than 5.
• If reciprocals are in play, correctly flipping them back to positive exponents can help visualize which is greater.

In exercises like these, restating the fractions back into whole numbers might also give clarity. So, when you convert reciprocals back and forth, ensure your solutions maintain consistency and sound logic for accurate comparisons.