Problem 87 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. $$\frac{8}{9 y^{2}}+\frac{1}{6 y^{4}}$$

Step-by-Step Solution

Answer

\( \frac{8}{9y^2} + \frac{1}{6y^4} = \frac{16y^2 + 3}{18y^4} \).

Step 1: Identify a Common Denominator

To add fractions, they must have a common denominator. Observe the fractions \( \frac{8}{9y^2} \) and \( \frac{1}{6y^4} \). The denominators are \(9y^2\) and \(6y^4\). The least common multiple (LCM) of these denominators is \(18y^4\).

Step 2: Create Equivalent Fractions

Convert each fraction to an equivalent fraction with the common denominator 18\(y^4\).- For \( \frac{8}{9y^2} \), multiply the numerator and the denominator by 2\(y^2\) to get \( \frac{16y^2}{18y^4} \).- For \( \frac{1}{6y^4} \), multiply the numerator and the denominator by 3 to get \( \frac{3}{18y^4} \).

Step 3: Add the Fractions

Now that the fractions have a common denominator, you can add them: \( \frac{16y^2}{18y^4} + \frac{3}{18y^4} = \frac{16y^2 + 3}{18y^4} \).

Step 4: Simplify the Expression

Check if the result can be simplified further. The expression \( \frac{16y^2 + 3}{18y^4} \) has no common factors between the numerator and the denominator that can be canceled, so the fraction is in its simplest form.

Key Concepts

Least Common Denominator

When adding or subtracting fractions, a crucial step is to find a common denominator. This makes it possible to combine the fractions into a single, unified expression. In our case, the task is to add \( \frac{8}{9y^2} \) and \( \frac{1}{6y^4} \). Each of these fractions has a different denominator, which makes finding the Least Common Denominator (LCD) necessary.

To find the LCD, we first need to determine the least common multiple (LCM) of the denominators. Here, the denominators are \( 9y^2 \) and \( 6y^4 \).

• We break down the constants and the variables:
• 9 = 3^2 and 6 = 2 \times 3.
• Gather common variables and powers: \( y^2 \) and \( y^4 \).

To combine everything, choose the highest power of each factor present, giving us the LCM of 18 and the variable part as \( y^4 \). Thus, the least common denominator is \( 18y^4 \).

This step ensures that both fractions will have the same denominator, making it possible to add them directly.

Equivalent Fractions

Next, it's necessary to convert each original fraction to an equivalent fraction that shares the least common denominator we found earlier, \( 18y^4 \). Equivalent fractions are different representations of the same value. We achieve this by adjusting both the numerator and the denominator by the same factor.

• For the fraction \( \frac{8}{9y^2} \), multiply both the numerator and the denominator by \( 2y^2 \):
• This yields \( \frac{16y^2}{18y^4} \).
• Similarly for \( \frac{1}{6y^4} \), multiply by 3:
• This gives us \( \frac{3}{18y^4} \).

By transforming each fraction to have a common denominator, they become compatible for addition. Each equivalent fraction represents the same quantity as the original, just in a different form that facilitates the operation.

Simplifying Fractions

After creating equivalent fractions with a common denominator and performing the addition, simplifying them if possible is the final step. Begin with the sum \( \frac{16y^2 + 3}{18y^4} \). Simplifying a fraction means reducing it to its simplest form without changing its value.

Check if the numerator and the denominator have any common factors. In this case, they do not, as the numerator components \( 16y^2 \) and 3 share no factors, and neither do they share factors with the denominator \( 18y^4 \).

Thus, \( \frac{16y^2 + 3}{18y^4} \) is already in its simplest form. Remember:

• Simplifying makes fractions easier to understand and work with.
• When no further common factors exist, the fraction is fully simplified.

Properly simplifying ensures clarity and preciseness in mathematical communication.