Problem 75 - Algebra for College Students

Question

For Problems \(75-84\), express each of the following as a single fraction involving positive exponents only. \(x^{-2}+x^{-3}\)

Step-by-Step Solution

Answer

\(\frac{x + 1}{x^3}\)

Step 1: Identify the Terms and Common Denominator

The expression given is \(x^{-2} + x^{-3}\). Both terms have different exponents. To combine them, we need to find a common denominator. Since both terms involve powers of \(x\), the common denominator is \(x^3\).

Step 2: Rewrite Each Term with the Common Denominator

Rewrite \(x^{-2}\) with the common denominator \(x^3\). \[x^{-2} = \frac{1}{x^2} = \frac{x}{x^3}\] Next, rewrite \(x^{-3}\) with the common denominator \(x^3\). \[x^{-3} = \frac{1}{x^3}\]

Step 3: Combine the Fractions into a Single Fraction

Add the rewritten fractions: \(\frac{x}{x^3} + \frac{1}{x^3}\). Since they have the same denominator, combine them: \[\frac{x + 1}{x^3}\]

Step 4: Simplify the Fraction

The fraction \(\frac{x + 1}{x^3}\) is already simplified and involves only positive exponents, meeting the requirements of the problem.

Key Concepts

Exponents

Exponents are a fundamental part of algebra and appear frequently in mathematical expressions. When you see a term like \(x^{-2}\), the exponent (in this case, \(-2\)) indicates how many times the base (\(x\)) is needed to be multiplied by itself. With negative exponents, the situation is a bit different. Negative exponents indicate that the base is on the wrong side of a fraction. Think of negative exponents as a way to express division rather than multiplication. For example:

• \(x^{-2} = \frac{1}{x^2}\)
• \(x^{-3} = \frac{1}{x^3}\)

This means that the base \(x\) should be moved to the denominator of a fraction. Understanding this concept can simplify the task of working with algebra fractions and will help in rewriting terms when necessary.

Combining Fractions

Combining fractions involves finding a common denominator, which allows us to add or subtract fractions easily. Here, the task is to express the given algebraic terms as a single fraction. The expression you've got is \(x^{-2} + x^{-3}\). Each term is a fraction, after you account for the negative exponents. The two denominators involved are \(x^2\) and \(x^3\); thus, the common denominator is \(x^3\). Why \(x^3\)? It's the least common multiple of the two denominators. To combine fractions:

• Convert each fraction to have \(x^3\) as the denominator: \(x^{-2} = \frac{x}{x^3}\) and \(x^{-3} = \frac{1}{x^3}\)
• Now that they have the same denominator, you can add the numerators: \(\frac{x}{x^3} + \frac{1}{x^3} = \frac{x + 1}{x^3}\)

This method simplifies combining fractions and is applicable to various algebra problems.

Simplifying Expressions

Simplifying expressions is the process of making an algebra expression easier to understand or solve by reducing it to its simplest form. In the case of the obtained fraction \(\frac{x + 1}{x^3}\), it appears challenging, but it is already in its simplest form.To simplify expressions:

• Ensure all exponents are positive, as required by the problem.
• Check if there are any common factors in the numerator and the denominator that can be further simplified. For this expression, there are none.
• Ensure the expression appears in its most compact form. Here, \(\frac{x+1}{x^3}\) is the simplest version.

Understanding how to simplify expressions helps avoid unnecessarily complicated solutions and makes it easier to identify the core components of an algebra problem.