Add the following.
$\frac{5}{4 y^3}+\frac{15}{4 y^3}= \square$ (Simplify your answer.)
Answer (1)
Add the numerators of the rational expressions: 5 + 15 = 20 .
Write the sum as a single rational expression: 4 y 3 20 .
Simplify the rational expression by dividing the numerator and the denominator by 4: 4 y 3 ÷ 4 20 ÷ 4 = y 3 5 .
The simplified expression is y 3 5 .
Explanation
Adding Rational Expressions We are asked to add two rational expressions: 4 y 3 5 and 4 y 3 15 . Since they have the same denominator, we can add the numerators directly.
Summing Numerators The sum of the numerators is 5 + 15 = 20 . So, we have 4 y 3 20 .
Simplifying the Fraction Now we simplify the fraction. We can divide both the numerator and the denominator by their greatest common divisor, which is 4. Thus, 4 y 3 20 = 4 y 3 ÷ 4 20 ÷ 4 = y 3 5 .
Examples
Understanding how to add and simplify rational expressions is crucial in many areas, such as electrical engineering when dealing with impedances in circuits, or in physics when analyzing wave phenomena. For example, when calculating the total impedance of two parallel impedances in an AC circuit, you often end up adding rational expressions. Simplifying these expressions allows engineers to efficiently design and analyze circuits, ensuring optimal performance and avoiding resonance issues. This skill is also fundamental in more advanced mathematical studies, providing a basis for calculus and differential equations.
