Problem 86 - Elementary Algebra

Question

For Problems \(75-92\), perform the indicated operations and express your answers in reduced form. $$ \frac{-8}{5 x^{3}}+\frac{10}{3 x^{3}} $$

Step-by-Step Solution

Answer

\(\frac{26}{15x^3}\)

Step 1: Identify the Common Denominator

In order to add or subtract fractions, they must have the same denominator. Here, both fractions have denominators involving \(x^3\), so the common denominator will be \(15x^3\), which is the least common multiple of 5 and 3 multiplied by \(x^3\).

Step 2: Rewrite Each Fraction with the Common Denominator

Rewrite each fraction so that they have the common denominator \(15x^3\). The first fraction \(\frac{-8}{5x^3}\) is equivalent to \(\frac{-8 \times 3}{5x^3 \times 3} = \frac{-24}{15x^3}\). The second fraction \(\frac{10}{3x^3}\) is equivalent to \(\frac{10 \times 5}{3x^3 \times 5} = \frac{50}{15x^3}\).

Step 3: Combine the Fractions

Now that both fractions have the same denominator, we can combine them by adding the numerators: \(\frac{-24}{15x^3} + \frac{50}{15x^3} = \frac{-24 + 50}{15x^3} = \frac{26}{15x^3}\).

Step 4: Simplify if Possible

Examine the final fraction \(\frac{26}{15x^3}\) to see if it can be simplified. In this case, 26 and 15 have no common factors other than 1, so this fraction is already in its simplest form.

Key Concepts

Common Denominator

When dealing with fraction operations, such as addition or subtraction, you'll need a common denominator. Let's break this down. A common denominator is simply a shared multiple of the denominators in your fractions. By finding a common denominator, you group the fractions under a single denominator, making it easier to perform operations.
In this exercise, the fractions are \(\frac{-8}{5x^3}\) and \(\frac{10}{3x^3}\). Both have the variable \(x^3\) in the denominator, but different numeric factors - 5 and 3. For such cases, find the least common multiple (LCM) of these numbers to get the common denominator.

• The LCM of 5 and 3 is 15.
• Our common denominator is \(15x^3\), incorporating the \(x^3\) variable from each fraction.

Using the common denominator, you can now adjust each fraction to this new denominator, enabling you to easily add or subtract.

Simplifying Algebraic Fractions

Once you've aligned your fractions to have a common denominator, the next step is simplifying them whenever possible. Simplifying fractions involves expressing them in their lowest terms, meaning the numerator and the denominator have no common factors other than 1.
Let's look at the fractions \(\frac{-24}{15x^3}\) and \(\frac{50}{15x^3}\) created in our solution:

• Both fractions are over a common denominator of \(15x^3\).
• The numerators are summarized by combining: \(-24 + 50 = 26\), forming the fraction \(\frac{26}{15x^3}\).

Inspecting \(\frac{26}{15x^3}\), the numbers 26 and 15 don't share any common factors apart from 1. Thus, this fraction is already fully simplified.

Rational Expressions

Rational expressions are essentially fractions where both the numerator and the denominator are algebraic expressions, often involving variables. Mastering rational expressions is crucial because they are pervasive in algebra and calculus.
In our original problem, both expressions, \(\frac{-8}{5x^3}\) and \(\frac{10}{3x^3}\), are rational expressions. They don't just have simple constants; they include the variable \(x^3\) in their denominators.
What makes rational expressions unique is the ability to transform and manipulate them, such as:

• Finding a common denominator, as we've done.
• Simplifying complex expressions into simpler forms.
• Equating and solving for values of variables within these expressions.

This ability to work seamlessly with rational expressions is essential not just for solving problems like this, but for understanding deeper math concepts.