Problem 47 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{7}{9 x y^{3}}-\frac{4}{3 x}+\frac{5}{2 y^{2}} $$

Step-by-Step Solution

Answer

\(\frac{14 - 24y^3 + 45x}{18xy^3}\)

Step 1: Finding the Least Common Denominator (LCD)

First, we need to identify the common denominator for all the fractions. The denominators are \(9xy^3\), \(3x\), and \(2y^2\). The least common multiple of these terms is \(18xy^3\). This will be our new common denominator.

Step 2: Rewrite Each Fraction with the LCD

For each fraction, convert it to have the denominator \(18xy^3\). - For \(\frac{7}{9xy^3}\), multiply both the numerator and denominator by 2: \[\frac{7 \times 2}{9xy^3 \times 2} = \frac{14}{18xy^3}\].- For \(\frac{4}{3x}\), multiply both the numerator and denominator by \(6y^3\): \[\frac{4 \times 6y^3}{3x \times 6y^3} = \frac{24y^3}{18xy^3}\].- For \(\frac{5}{2y^2}\), multiply both the numerator and denominator by \(9x\): \[\frac{5 \times 9x}{2y^2 \times 9x} = \frac{45x}{18xy^3}\].

Step 3: Combine the Fractions

Now that all fractions have the same denominator, combine them: \[\frac{14}{18xy^3} - \frac{24y^3}{18xy^3} + \frac{45x}{18xy^3} = \frac{14 - 24y^3 + 45x}{18xy^3}\].

Step 4: Simplify the Expression

Check if the combined numerator can be simplified or factored further. In this case, the expression \(14 - 24y^3 + 45x\) has no common simple factors, so the fraction is already in its simplest form. Thus, the simplified expression is \[\frac{14 - 24y^3 + 45x}{18xy^3}\].

Key Concepts

Least Common Denominator

When working with rational expressions, finding a common denominator is essential for combining fractions. The least common denominator (LCD) is the smallest expression that each of the denominators can divide without leaving a remainder. In this exercise, we have denominators of \(9xy^3\), \(3x\), and \(2y^2\).

To find the LCD, we identify all the unique factors among the denominators. Let's break it down:

• \(9xy^3\) has the factors \(9\), \(x\), and \(y^3\).
• \(3x\) has the factors \(3\) and \(x\).
• \(2y^2\) has the factors \(2\) and \(y^2\).

The LCD must contain the highest power of each factor present in any denominator. This means the LCD will be \(18xy^3\). Here, \(18\) is obtained from the least common multiple of \(9, 3,\) and \(2\), and \(xy^3\) is used as it is the maximum degree found in the denominators.

Understanding how to find the LCD is crucial because it sets the stage for rewriting each fraction so they can be easily combined later.

Simplifying Fractions

Simplifying fractions is an important step before combined rational expressions. It involves reducing each fraction to its smallest form, but more importantly, adjusting them to share a common denominator. In this exercise, once we find the least common denominator, each fraction must be rewritten in terms of this LCD.

The original fractions are:

• \(\frac{7}{9xy^3}\), which is multiplied by 2 to become \(\frac{14}{18xy^3}\).
• \(\frac{4}{3x}\), which is multiplied by \(6y^3\) to become \(\frac{24y^3}{18xy^3}\).
• \(\frac{5}{2y^2}\), which is multiplied by \(9x\) to become \(\frac{45x}{18xy^3}\).

This step ensures compatibility among the fractions, allowing them to be directly added or subtracted. Simplifying fractions this way maintains the integrity of the values while setting up for easy combination. The multipliers used here are chosen to transform each denominator into the least common denominator we determined earlier.

Combining Fractions

After fractions are rewritten with a common denominator, they can be combined. This involves adding or subtracting the numerators while maintaining the denominator. Since all fractions now have the denominator \(18xy^3\), the next step is straightforward: combine the numerators.

We perform the following operation:\[\frac{14}{18xy^3} - \frac{24y^3}{18xy^3} + \frac{45x}{18xy^3}\]This combines into:\[\frac{14 - 24y^3 + 45x}{18xy^3}\]Once the fractions are combined, always consider checking for simplification opportunities. In this case, the expression \(14 - 24y^3 + 45x\) has no common factors, indicating the fraction is already in its simplest form.

This final step allows for the consolidation of terms, giving a clear and concise expression as the result.