Problem 39 - Algebra for College Students

Question

Simplify each numerical expression. \(\left(\frac{1}{3}\right)^{-1}-\left(\frac{2}{5}\right)^{-1}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{2}\).

Step 1: Understand the expression

The expression to simplify is \(\left(\frac{1}{3}\right)^{-1} - \left(\frac{2}{5}\right)^{-1}\). The negative exponent indicates that we take the reciprocal of the fraction, so \(\left(\frac{1}{3}\right)^{-1}\) becomes \(3\) and \(\left(\frac{2}{5}\right)^{-1}\) becomes \(\frac{5}{2}\). Therefore, the expression simplifies to \(3 - \frac{5}{2}\).

Step 2: Convert whole number to fraction

In order to perform the subtraction \(3 - \frac{5}{2}\), convert \(3\) to a fraction with the same denominator as \(\frac{5}{2}\). Thus, \(3\) becomes \(\frac{6}{2}\), since \(3 \times \frac{2}{2} = \frac{6}{2}\).

Step 3: Subtract the fractions

Subtract \(\frac{5}{2}\) from \(\frac{6}{2}\): \(\frac{6}{2} - \frac{5}{2} = \frac{6-5}{2} = \frac{1}{2}\). The expression simplifies to \(\frac{1}{2}\).

Key Concepts

Negative Exponents

Negative exponents might seem confusing at first, but they follow a simple rule. A negative exponent essentially tells you to take the reciprocal of the base. This means that for a fraction like \( \left( \frac{1}{3} \right)^{-1} \), you would flip the fraction to become \( 3 \), or \( \frac{3}{1} \). It is important to note that this rule applies to all numbers, not just fractions.

For instance, \( a^{-n} = \frac{1}{a^n} \). Similarly, \( \left( \frac{x}{y} \right)^{-n} = \left( \frac{y}{x} \right)^{n} \). In our example from the exercise, applying this rule to \( \left( \frac{2}{5} \right)^{-1} \) gives us \( \frac{5}{2} \).

To summarize, a negative exponent instructs you to deal with the reciprocal, making the expression often easier to handle for further operations.

Fraction Operations

Understanding how to work with fractions is crucial, especially when dealing with expressions requiring subtraction or addition. When fractions have different denominators, the first step is to find a common denominator. However, if they share a denominator, you can directly proceed with the operation.

In our exercise, after converting the whole number 3 into the fraction \( \frac{6}{2} \), both fractions have the same denominator. This makes subtraction straightforward: \( \frac{6}{2} - \frac{5}{2} \).

Remember that when adding or subtracting fractions, only the numerators change; the denominator remains constant. You simply perform the arithmetic operation on the numerators:

• \( \frac{6-5}{2} = \frac{1}{2} \)

This rule simplifies the process and ensures you end with simplified, precise results.

Simplifying Expressions

Simplifying expressions is about making them easier to understand or work with, often by condensing them to fewer terms or easier numbers. This can involve a range of techniques, from factorization to operations like the ones in our exercise.

To simplify fractions with negative exponents as in our exercise, the first step was dealing with the negative exponents—this transformed the expression into something manageable: \( 3 - \frac{5}{2} \).

Then, converting whole numbers to fractions when necessary allowed us to combine terms using common denominators. The goal was to reduce the expression to its simplest form, \( \frac{1}{2} \), by performing clear operations step-by-step.

• Identify simpler equivalent expressions.
• Use fundamental arithmetic skills to combine expressions.
• Apply basic algebraic principles to ensure accuracy.

Following these steps aids in achieving a simplified, clear result.