Problem 83 - Algebra for College Students

Question

Express each of the following as a single fraction involving positive exponents only. \(2 x^{-1}-3 x^{-2}\)

Step-by-Step Solution

Answer

\(\frac{2x - 3}{x^2}\).

Step 1: Rewrite with Positive Exponents

First, rewrite the expression using positive exponents. Recall that negative exponents indicate the reciprocal. Therefore, \(x^{-1} = \frac{1}{x}\) and \(x^{-2} = \frac{1}{x^2}\). The expression \(2x^{-1} - 3x^{-2}\) becomes \(\frac{2}{x} - \frac{3}{x^2}\).

Step 2: Find the Common Denominator

Identify the common denominator for the fractions \(\frac{2}{x}\) and \(\frac{3}{x^2}\). The least common denominator (LCD) is \(x^2\).

Step 3: Convert Fractions to the Common Denominator

Rewrite each fraction with a denominator of \(x^2\). Multiply the numerator and denominator of \(\frac{2}{x}\) by \(x\): \(\frac{2}{x} = \frac{2x}{x^2}\). The fraction \(\frac{3}{x^2}\) already has \(x^2\) as its denominator.

Step 4: Combine the Fractions

Now, combine the fractions. \(\frac{2x}{x^2} - \frac{3}{x^2} = \frac{2x - 3}{x^2}\).

Step 5: Simplify if Necessary

Check if the expression \(\frac{2x - 3}{x^2}\) can be simplified further. In this case, it is already in its simplest form.

Key Concepts

Negative Exponents

Understanding negative exponents is a key concept in algebra and mathematics in general. When you see a negative exponent like in the term \(x^{-1}\), it tells you to take the reciprocal of the base and make the exponent positive.
For instance, \(x^{-1}\) is the same as \(\frac{1}{x}\). Similarly, \(x^{-2}\) becomes \(\frac{1}{x^2}\).
This transformation from a negative to a positive exponent is crucial because it simplifies equations and expressions, making them easier to work with and understand.
In our example, taking you from \(2 x^{-1} - 3 x^{-2}\) to \(\frac{2}{x} - \frac{3}{x^2}\) gives you a clear view of how things change when negative exponents are involved.

Common Denominator

Finding a common denominator is an essential step when you are working with fractions.
To add or subtract fractions, they need to have the same denominator. This denominator is called the least common denominator (LCD).
Think of it as extending the number line to ensure all the fractions can be properly aligned.

• In our case, the fractions are \(\frac{2}{x}\) and \(\frac{3}{x^2}\).
• The denominators here are \(x\) and \(x^2\). The LCD, therefore, is \(x^2\).

To make the denominators common, adjust the fractions while keeping their values the same.
This makes combining them straightforward and leaves you with cleaner, unified expressions to solve.

Simplifying Fractions

Simplifying fractions involves making them as straightforward as possible by reducing or adjusting components without altering their value. This is important to ensure you’re always working with the simplest form of an expression.

• After finding a common denominator and adjusting fractions accordingly, you will often need to combine them, as seen when combining \(\frac{2x}{x^2}\) and \(\frac{3}{x^2}\).
• Here, the operation results in \(\frac{2x - 3}{x^2}\).

This fraction is already simplified as there are no common factors between the numerator \(2x - 3\) and the denominator \(x^2\).
Simplification ensures that the expression is in its cleanest form, which helps in solving equations smoothly and efficiently.

Positive Exponents

Working with positive exponents turns complicated expressions into more manageable ones.
Positive exponents like those resulting from converting negative exponents give clarity.
They are straightforward to handle because they show straightforward multiplication of the base.

• In the earlier steps, converting \(x^{-1}\) and \(x^{-2}\) to \(\frac{1}{x}\) and \(\frac{1}{x^2}\), respectively, transforms the original equation into simpler terms with positive exponents.
• Positive exponents represent real-world scaling and proportions directly.

In algebra, expressing problems with positive exponents is like tidying up a room—it makes everything easier to see and handle.
This clarity makes subsequent operations, like addition and subtraction of fractions, hassle-free and precise.