Problem 87 - Elementary Algebra

Question

For Problems \(75-92\), perform the indicated operations and express your answers in reduced form. $$ \frac{11}{a^{2}}-\frac{14}{b^{2}} $$

Step-by-Step Solution

Answer

\(\frac{11b^2 - 14a^2}{a^2b^2}\) is the reduced form.

Step 1: Identify the Operation

The problem involves subtracting two fractions: \( \frac{11}{a^2} \) and \( \frac{14}{b^2} \). Our goal is to combine these fractions using subtraction.

Step 2: Find a Common Denominator

The denominators of the fractions are \( a^2 \) and \( b^2 \). The least common denominator (LCD) is the product of these distinct denominators, which is \( a^2b^2 \).

Step 3: Rewrite Each Fraction

To rewrite both fractions with the common denominator, multiply each fraction by the other fraction's denominator over itself.\[\frac{11}{a^2} = \frac{11 \cdot b^2}{a^2b^2}\]\[\frac{14}{b^2} = \frac{14 \cdot a^2}{a^2b^2}\]

Step 4: Subtract the Fractions

Now subtract the two fractions that have the same denominator:\[\frac{11b^2}{a^2b^2} - \frac{14a^2}{a^2b^2} = \frac{11b^2 - 14a^2}{a^2b^2}\]

Step 5: Simplify the Result

Since there are no common factors between the numerator (\(11b^2 - 14a^2\)) and the denominator (\(a^2b^2\)), this fraction is already in its simplest form.

Key Concepts

Common Denominator

A common denominator is a shared multiple of the denominators of two or more fractions. Think of it as a common ground that allows us to compare or combine fractions without altering their values. In the given exercise, we are working with the fractions \( \frac{11}{a^2} \) and \( \frac{14}{b^2} \). To subtract them, they need to have the same denominator.
To find a common denominator, you take the least common multiple (LCM) of the denominators involved. When the denominators are distinct, as in this case with \( a^2 \) and \( b^2 \), their LCM is simply their product: \( a^2b^2 \). This common denominator lets us rewrite both fractions such that they can be subtracted from one another.
By ensuring that fractions have this unified base, we can easily perform arithmetic operations and maintain mathematical accuracy. Thus, finding a common denominator is crucial when working with algebraic fractions.

Subtracting Fractions

Subtracting fractions involves aligning them to the same common denominator. Once they share a common denominator, you can directly subtract the numerators while keeping the denominator unchanged. This process is straightforward but requires careful handling of the signs and the expressions in the numerators.
In our example, after converting \( \frac{11}{a^2} \) and \( \frac{14}{b^2} \) to \( \frac{11b^2}{a^2b^2} \) and \( \frac{14a^2}{a^2b^2} \) respectively, we just subtract the numerators:

• The subtraction becomes: \( 11b^2 - 14a^2 \)
• The shared denominator remains: \( a^2b^2 \)

The result is expressed as \( \frac{11b^2 - 14a^2}{a^2b^2} \). This step ensures that the subtraction process is aligned correctly for the fractions involved, preparing them for potential simplification.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. For numeric fractions, you might perform division to simplify. However, with algebraic fractions, you check and factor expressions.
With the expression \( \frac{11b^2 - 14a^2}{a^2b^2} \), we examine whether the numerator \( 11b^2 - 14a^2 \) shares any common factors with the denominator \( a^2b^2 \). Upon inspection:

• The terms in the numerator and the denominator have no common factors.
• Thus, this fraction is already in its simplest form.

Simplification helps in understanding the expression better and also ensures that any further calculations use the most reduced terms possible. In algebra, this clarity is crucial for maintaining accuracy in larger problems or equations.