Problem 88 - Elementary Algebra

Question

For Problems \(75-92\), perform the indicated operations and express your answers in reduced form. $$ \frac{9}{x^{2}}+\frac{8}{y^{2}} $$

Step-by-Step Solution

Answer

\( \frac{9y^2 + 8x^2}{x^2y^2} \)

Step 1: Identify the Like Denominators

In the given expression \( \frac{9}{x^{2}} + \frac{8}{y^{2}} \), the denominators \( x^2 \) and \( y^2 \) are different. We cannot add the fractions directly because their denominators are not the same.

Step 2: Find the Least Common Denominator (LCD)

The least common denominator of \( x^2 \) and \( y^2 \) is their product \( x^2y^2 \). This is because they are distinct with no common factors.

Step 3: Rewrite Fractions with the LCD

Convert each fraction to have the common denominator \( x^2y^2 \):- For \( \frac{9}{x^2} \), multiply numerator and denominator by \( y^2 \), resulting in \( \frac{9y^2}{x^2y^2} \).- For \( \frac{8}{y^2} \), multiply the numerator and denominator by \( x^2 \), yielding \( \frac{8x^2}{x^2y^2} \).

Step 4: Add the Fractions

Now that both fractions have a common denominator, add them together:\[\frac{9y^2}{x^2y^2} + \frac{8x^2}{x^2y^2} = \frac{9y^2 + 8x^2}{x^2y^2}\]

Step 5: Simplify the Final Expression

Check if the numerator \( 9y^2 + 8x^2 \) can be factored or the fraction can be simplified further. In this case, it cannot be simplified further, so \( \frac{9y^2 + 8x^2}{x^2y^2} \) is in its simplest form.

Key Concepts

Least Common Denominator (LCD)

To successfully work with algebraic fractions, finding the least common denominator (LCD) is crucial. It allows us to combine fractions by rewriting each with a shared denominator. The LCD is the smallest expression divisible by each of the original denominators.
In our exercise, we have fractions with denominators \( x^2 \) and \( y^2 \). Each is distinct, without any common factors, making the LCD simply the product of the two: \( x^2y^2 \).
Important steps to find the LCD:

• Identify all denominators in the given expressions.
• Factorize each denominator, if possible.
• Multiply all distinct factors to find the LCD.

Understanding the concept of the LCD will significantly ease the process of combining and simplifying algebraic fractions.

Adding Fractions

Once each fraction has been rewritten to have the same denominator, we can proceed to addition. The numerator of each fraction needs to be adjusted appropriately to account for the new common denominator.
Here's how you add fractions with an LCD:

• Ensure each fraction has the LCD. Rewrite them by multiplying both the numerator and denominator by the factors needed to obtain the LCD.
• In our example, \( \frac{9}{x^2} \) becomes \( \frac{9y^2}{x^2y^2} \) and \( \frac{8}{y^2} \) becomes \( \frac{8x^2}{x^2y^2} \).
• Add the numerators while keeping the common denominator the same. The result is \( \frac{9y^2 + 8x^2}{x^2y^2} \).

These steps make sure the addition is handled correctly, keeping the mathematical expressions accurate.

Simplification of Expressions

Simplifying expressions composes the final step in many algebraic problems. Once you have combined the fractions using the LCD, it's essential to reduce the expression to its simplest form if possible.
Here's what to keep in mind for simplification:

• Combine like terms in the numerator if there are any.
• Factorize the numerator if possible to see if it shares any factors with the denominator.
• If no further simplification is possible, as in our example with \( \frac{9y^2 + 8x^2}{x^2y^2} \), where no common factors exist, you have your final answer.

This process ensures clarity and precision, making the math more manageable and results straightforward to work with.