Problem 48 - Precalculus Mathematics for Calculus

Question

Perform the addition or subtraction and simplify. $$\frac{2}{a^{2}}-\frac{3}{a b}+\frac{4}{b^{2}}$$

Step-by-Step Solution

Answer

\(\frac{2b^2 - 3ab + 4a^2}{a^2b^2}\) (as simplified as possible).

Step 1: Identify the Least Common Denominator (LCD)

First, observe the denominators: \(a^2\), \(ab\), and \(b^2\). To perform addition or subtraction of fractions, the denominators need to be the same. The least common denominator for these expressions is \(a^2b^2\). This is because \(a^2b^2\) is the smallest expression that each of the individual denominators can divide into without leaving a remainder.

Step 2: Express Each Fraction with the LCD

Convert each fraction so that it has the LCD, \(a^2b^2\), as its denominator. - For \(\frac{2}{a^2}\), multiply the numerator and denominator by \(b^2\) to get \(\frac{2b^2}{a^2b^2}\).- For \(\frac{3}{ab}\), multiply the numerator and denominator by \(ab\) to obtain \(\frac{3ab}{a^2b^2}\).- For \(\frac{4}{b^2}\), multiply the numerator and denominator by \(a^2\) to get \(\frac{4a^2}{a^2b^2}\).

Step 3: Combine the Fractions

Now that all the fractions have a common denominator, combine them into a single fraction:\[\frac{2b^2}{a^2b^2} - \frac{3ab}{a^2b^2} + \frac{4a^2}{a^2b^2} = \frac{2b^2 - 3ab + 4a^2}{a^2b^2}.\]

Step 4: Simplify the Numerator

Examine if the numerator can be simplified further. In this case, the expression \(2b^2 - 3ab + 4a^2\) does not factor to a simpler form easily or completely, so we leave this as is.

Key Concepts

Least Common Denominator

When dealing with rational expressions, especially in addition and subtraction, finding the Least Common Denominator (LCD) is a crucial step. The LCD is the smallest expression that all denominators can divide into without a remainder.
For example, consider fractions with denominators such as \(a^2\), \(ab\), and \(b^2\). To find the LCD, identify the highest powers of each variable present in all denominators:

• \(a^2\) for the variable \(a\)
• \(b^2\) for the variable \(b\)

Putting them together forms the LCD: \(a^2b^2\). Finding the LCD allows you to convert each fraction so they share a common denominator, enabling straightforward addition or subtraction.

Fraction Addition

Once each fraction is expressed with the same denominator, you can easily add or subtract them. This process requires uniform denominators to maintain the integrity of the operation. In our example, the fractions become:

• \(\frac{2b^2}{a^2b^2}\)
• \(\frac{3ab}{a^2b^2}\)
• \(\frac{4a^2}{a^2b^2}\)

Now, all terms share the denominator \(a^2b^2\). Simply combine the numerators to form a new fraction:
\[ \frac{2b^2 - 3ab + 4a^2}{a^2b^2} \]
By having a shared denominator, the process of addition or subtraction is simplified. This helps to focus solely on the numerators.

Fraction Simplification

After combining the fractions, the last step is to simplify the resultant rational expression. Look at the numerator and check if it can be factored or reduced further. Simplifying the expression reduces it to its simplest terms.
In our exercise, the numerator \(2b^2 - 3ab + 4a^2\) does not factor easily. In such cases, if the numerator cannot be reduced or factored succinctly, leave the expression as is.
Simplification is essential in mathematics as it helps to make expressions and equations more manageable. Assess each expression critically to ensure maximum simplification, improving clarity and function of the expression.