Problem 36 - Algebra for College Students

Question

Add or subtract the rational expressions as indicated. Be sure to express your answers in simplest form. $$ \frac{7}{12 a^{2}}-\frac{5}{16 a} $$

Step-by-Step Solution

Answer

\(\frac{28 - 15a}{48a^2}\)

Step 1: Identify the Least Common Denominator (LCD)

The denominators of the rational expressions are \(12a^2\) and \(16a\). To find the Least Common Denominator, identify the least common multiple of the numbers 12 and 16, which is 48. For the variables, the least common multiple of \(a^2\) and \(a\) is \(a^2\). Therefore, the LCD is \(48a^2\).

Step 2: Rewrite Each Fraction with the LCD

To rewrite the first fraction with the LCD, multiply both the numerator and the denominator by 4 to get \(\frac{7}{12a^2} = \frac{28}{48a^2}\). For the second fraction, multiply both the numerator and the denominator by 3a to get \(\frac{5}{16a} = \frac{15a}{48a^2}\).

Step 3: Perform the Subtraction

Now that both fractions have the same denominator, subtract the numerators: \(\frac{28}{48a^2} - \frac{15a}{48a^2} = \frac{28 - 15a}{48a^2}\).

Step 4: Simplify the Expression

Since \(28 - 15a\) and \(48a^2\) have no common factors, the rational expression is already in its simplest form. Thus, the final answer is \(\frac{28 - 15a}{48a^2}\).

Key Concepts

Least Common Denominator

When working with rational expressions, finding the Least Common Denominator (LCD) is a crucial step. The LCD lets us combine fractions by giving them the same denominator. To find the LCD, follow these steps:

• Identify the denominators of each rational expression. In this exercise, the denominators are \(12a^2\) and \(16a\).
• Determine the least common multiple (LCM) of the numerical coefficients. Here, the LCM of 12 and 16 is 48.
• Consider the variables involved. Compare the exponents of the variables and choose the highest exponent for each variable. The LCM of \(a^2\) and \(a\) is \(a^2\).

Combine these results to get the LCD: \(48a^2\). This common denominator allows you to rewrite both fractions so they can be added or subtracted easily.

Simplifying Expressions

Once the fractions share a common denominator, it's time to simplify the expressions. Simplifying makes calculations easier and your final answer clearer. Start by rewriting the fractions with the LCD.
For example:

• For \(\frac{7}{12a^2}\), multiply both the numerator and denominator by 4 to get \(\frac{28}{48a^2}\).
• For \(\frac{5}{16a}\), multiply both the numerator and denominator by 3a to achieve \(\frac{15a}{48a^2}\).

By creating equivalent fractions with the same denominator, we're set to perform subtraction. Simplifying ensures the calculation process is straightforward and lays the groundwork for further reduction if possible.

Subtracting Fractions

Once the fractions are expressed with a common denominator, the process of subtracting becomes seamless.

• Subtract the numerators while keeping the common denominator constant. So, from the example \(\frac{28}{48a^2} - \frac{15a}{48a^2} = \frac{28 - 15a}{48a^2}\).
• Always check if the resulting numerator and denominator have any common factors for further simplification. In this example, there are none, so the expression is already simplified.

Subtracting fractions involves aligning their denominators first. This consistency allows the subtraction of the numerators, facilitating the final solution. Always ensure your result is presented in its simplest form, making sure nothing more can be factored out easily.