Math Problem Statement

Perform the operation and express your answer as a single fraction in simplest form: \( \frac{1}{2x^2} + \frac{4x}{3} \).

Solution

To solve 12x2+4x3\frac{1}{2x^2} + \frac{4x}{3} as a single fraction in simplest form, we will follow these steps:


Step 1: Find a common denominator

The denominators are 2x22x^2 and 33.
The least common denominator (LCD) is: LCD=6x2\text{LCD} = 6x^2


Step 2: Adjust the fractions to have the LCD

  1. For 12x2\frac{1}{2x^2}, multiply the numerator and denominator by 3: 12x2=36x2\frac{1}{2x^2} = \frac{3}{6x^2}
  2. For 4x3\frac{4x}{3}, multiply the numerator and denominator by 2x22x^2: 4x3=8x36x2\frac{4x}{3} = \frac{8x^3}{6x^2}

Step 3: Combine the fractions

Now that both fractions have the same denominator 6x26x^2, add the numerators: 36x2+8x36x2=3+8x36x2\frac{3}{6x^2} + \frac{8x^3}{6x^2} = \frac{3 + 8x^3}{6x^2}


Final Answer:

The result expressed as a single fraction is: 8x3+36x2\frac{8x^3 + 3}{6x^2}

Let me know if you need more details or further explanation!


Related Questions:

  1. How do you find the least common denominator for more complex fractions?
  2. What are the steps to simplify fractions involving variables?
  3. How do you perform operations on fractions with polynomials in the numerator?
  4. How do you factorize the numerator and denominator of rational expressions?
  5. What is the process for reducing fractions to their simplest form?

Tip: Always check for common factors in the numerator and denominator to simplify further!

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Math Problem Analysis

Mathematical Concepts

Algebra
Fractions with variables
Least Common Denominator (LCD)
Simplification of rational expressions

Formulas

Finding the least common denominator: \( LCD = \text{LCM of denominators} \)
Addition of fractions: \( \frac{a}{c} + \frac{b}{c} = \frac{a + b}{c} \)

Theorems

Properties of rational expressions

Suitable Grade Level

Grades 9-11