In Exercises 33–68, add or subtract as indicated. (x+9)/10x^3 + 11/15x^2

Verified step by step guidance

1. Identify the least common denominator (LCD) for the fractions. The denominators are $10x^3$ and $15x^2$.
2. The LCD is the least common multiple of $10x^3$ and $15x^2$. Find the LCM by taking the highest power of each factor present in the denominators.
3. The LCM of the numerical coefficients 10 and 15 is 30. The LCM of $x^3$ and $x^2$ is $x^3$. Therefore, the LCD is $30x^3$.
4. Rewrite each fraction with the LCD as the new denominator. Multiply the numerator and denominator of the first fraction by 3, and the second fraction by 2, to achieve the common denominator.
5. Combine the fractions by adding the numerators over the common denominator. Simplify the resulting expression if possible.

Key Concepts

Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Understanding how to manipulate these expressions, including addition and subtraction, is crucial for solving problems involving them. This often requires finding a common denominator to combine the fractions effectively.

Finding a Common Denominator

To add or subtract rational expressions, it is essential to find a common denominator. This involves determining the least common multiple (LCM) of the denominators involved. Once a common denominator is established, the numerators can be adjusted accordingly, allowing for the expressions to be combined.

Simplifying Rational Expressions

After performing operations on rational expressions, simplifying the result is important. This includes factoring the numerator and denominator to cancel out any common factors. Simplification helps in presenting the final answer in its most reduced form, making it easier to interpret and use in further calculations.