Problem 46 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{9}{14 x^{2} y}-\frac{4 x}{7 y^{2}} $$

Step-by-Step Solution

Answer

\(\frac{9y - 8x^3}{14x^2y^2}\)

Step 1: Find a Common Denominator

To add or subtract rational expressions, we must first find a common denominator. The denominators are \(14x^2y\) and \(7y^2\). The least common denominator (LCD) is determined by taking the highest power of each factor appearing in any of the denominators.1. The factors are \(14x^2y = 2 \times 7 \times x^2 \times y\).2. The factors for \(7y^2 = 7 \times y^2\).Thus, the LCD is \(14x^2y^2\).

Step 2: Rewrite Each Fraction with the Common Denominator

Rewrite each expression so that they have the common denominator \(14x^2y^2\).1. Multiply the numerator and denominator of \(\frac{9}{14x^2y}\) by \(y\) to get \(\frac{9y}{14x^2y^2}\).2. Multiply the numerator and denominator of \(\frac{4x}{7y^2}\) by \(2x^2\) to get \(\frac{8x^3}{14x^2y^2}\).

Step 3: Subtract the Numerators

Now that both fractions have the same denominator, subtract the numerators:\[\frac{9y}{14x^2y^2} - \frac{8x^3}{14x^2y^2} = \frac{9y - 8x^3}{14x^2y^2}\]

Step 4: Simplify the Result

The expression \(\frac{9y - 8x^3}{14x^2y^2}\) cannot be simplified further as there are no common factors between the numerator and the denominator.Thus, the final simplified form of the expression is \(\frac{9y - 8x^3}{14x^2y^2}\).

Key Concepts

Understanding Common Denominator

When we deal with rational expressions, a common denominator is crucial to adding or subtracting them. A rational expression is similar to a fraction, except it involves variables. To successfully perform operations such as addition or subtraction, the fractions need to share the same denominator.
In our example, we have two fractions: \(\frac{9}{14x^2y}\) and \(\frac{4x}{7y^2}\). Before subtracting them, we identify a common denominator.
This is akin to finding a common multiple of their denominators.

• The denominators \(14x^2y\) and \(7y^2\) break down into their prime factors: \(14x^2y = 2 \times 7 \times x^2 \times y\) and \(7y^2 = 7 \times y^2\).
• The common denominator is formed by taking the highest power of each factor that is present in the denominators. In this case, it is \(14x^2y^2\).

Once the common denominator \(14x^2y^2\) is found, it allows us to rewrite both fractions uniformly, moving us closer to subtraction.

Subtracting Fractions: The Method

With rational expressions now having a shared denominator, we can proceed to subtract the fractions. Just like numerical fractions, subtracting rational expressions involves only the numerators since the denominators are already the same.
In our example, the expressions \(\frac{9y}{14x^2y^2}\) and \(\frac{8x^3}{14x^2y^2}\) have a common denominator of \(14x^2y^2\).

• To subtract, simply compute \(9y - 8x^3\), placing the result over the same denominator: \(\frac{9y - 8x^3}{14x^2y^2}\).
• There’s no need to alter the denominator at this stage.The subtraction process is simplified because the denominator remains constant throughout.

This method ensures clarity and ease when working with complex rational expressions.

Simplifying Rational Expressions

Simplifying rational expressions is the final step, ensuring the expression is in its simplest form.
After performing operations such as addition or subtraction, it's beneficial to look for common factors in the numerator and denominator that can be canceled out.In our example, \(\frac{9y - 8x^3}{14x^2y^2}\), you would check if there are any factors common to both parts of the fraction.

• However, we find no common factors between \(9y - 8x^3\) and \(14x^2y^2\), confirming that the expression is already simplified.
• Always remember: simplification involves factorizing both the numerator and the denominator and canceling out any shared factors.

Being able to identify when an expression is fully simplified is vital, as it confirms the completeness of your work, ensuring clarity in the result.