Problem 18 - Algebra 1: Concepts and Skills

Question

SUBTRACTING RATIONAL EXPRESSIONS. Simplify the expression. $$ \frac{7 x}{x^{3}}-\frac{6 x}{x^{3}} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{1}{x^{2}}\).

Step 1: Identify the Like Terms

The given expressions have the same denominator, so they are 'like' fractions. This means that we can subtract the numerators directly.

Step 2: Subtract the Numerators

We subtract the two numerators, which are \(7x\) and \(6x\). This will give us \(7x - 6x\). This subtraction results in \(x\).

Step 3: Simplify the Fraction

Our fraction currently looks like this: \(\frac{x}{x^{3}}\). To further simplify, we can divide \(x\) in the numerator with one of the \(x\) in the \(x^{3}\). The result is: \(\frac{1}{x^{2}}\).

Key Concepts

Subtracting Fractions

When dealing with the subtraction of fractions, especially rational expressions, it’s crucial to first understand if the fractions have a common denominator. For the subtraction of fractions like \[ \frac{7x}{x^3} - \frac{6x}{x^3} \] the denominators are already the same, which simplifies the process significantly. This means you can directly subtract the numerators.
Breaking down the subtraction:

• Identify if the fractions are 'like', meaning they have the same denominator.
• Once confirmed, subtract the numerators as demonstrated:
  \[ 7x - 6x = x \]
• Keep the common denominator the same (in this case, \(x^3\)).

Dealing with like terms in fraction subtraction simplifies your work because you only need to focus on the numerators.

Simplifying Expressions

Simplifying expressions, especially rational ones, involves reducing fractions to their simplest form. After performing operations such as addition or subtraction, it's important to see if the resulting expression can be simplified.
For example, from the earlier step, we obtained \[ \frac{x}{x^3} \]. This can be simplified by:

• Canceling common factors in the numerator and denominator. Here, \(x\) in the numerator and one of the \(x\) terms in \(x^3\).
• This gives us: \[ \frac{1}{x^2} \].

Remember, simplifying fractions after operations ensures that you have the most reduced and manageable form of the expression, making further calculations easier if necessary.

Like Terms

Working with like terms means dealing with algebraic expressions having the same variables raised to the same power. This concept is indispensable when adding or subtracting terms in algebra.
In the context of \( \frac{7x}{x^3} - \frac{6x}{x^3} \):

• Both numerators, \(7x\) and \(6x\), are like terms because they both consist of \(x\) raised to the power of 1.
• Like terms enable direct subtraction, allowing simplification, as seen by \(7x - 6x = x\).

Handling like terms effectively permits clearer and more accurate manipulation of expressions. It’s a powerful tool to simplify complex expressions effortlessly.