Perform the indicated operation.
$\frac{3}{x}+\frac{5}{3 x^2}= \square$ (Simplify your answer)
Answer (1)
Find a common denominator for the two fractions, which is 3 x 2 .
Rewrite each fraction with the common denominator: x 3 = 3 x 2 9 x and 3 x 2 5 remains the same.
Add the two fractions: 3 x 2 9 x + 3 x 2 5 = 3 x 2 9 x + 5 .
The simplified expression is 3 x 2 9 x + 5 .
Explanation
Understanding the Problem We are asked to add two rational expressions: x 3 and 3 x 2 5 . Our goal is to simplify the result into a single fraction.
Finding the Common Denominator To add these fractions, we need a common denominator. The least common denominator (LCD) of x and 3 x 2 is 3 x 2 .
Rewriting Fractions We rewrite each fraction with the common denominator 3 x 2 . To do this, we multiply the numerator and denominator of the first fraction by 3 x : x 3 = x ( 3 x ) 3 ( 3 x ) = 3 x 2 9 x . The second fraction already has the common denominator: 3 x 2 5 .
Adding Fractions Now we add the two fractions: 3 x 2 9 x + 3 x 2 5 = 3 x 2 9 x + 5 .
Final Simplification We check if the resulting fraction can be simplified further. In this case, the numerator and denominator have no common factors, so the fraction is already in simplest form. Therefore, the simplified expression is 3 x 2 9 x + 5 .
Examples
Rational expressions are useful in many real-world applications, such as calculating rates, proportions, and changes in quantities. For example, if you are mixing a solution in a lab, you might use rational expressions to determine the concentration of a substance as you add more solvent. Similarly, in business, you can use rational expressions to analyze profit margins or cost-benefit ratios. Understanding how to add and simplify these expressions allows for efficient problem-solving in these scenarios.
