Subtract: 7 6 U 2 X − 2 9 U X 3 \frac{7}{6 U^2 X} - \frac{2}{9 U X^3} 6 U 2 X 7 − 9 U X 3 2

Mar 13, 2025 by ADMIN 96 views

**Subtracting Fractions with Variables: A Step-by-Step Guide**

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Understanding the Problem

When dealing with fractions that contain variables, it's essential to understand the rules for subtracting them. In this article, we'll focus on subtracting two fractions with variables, specifically the expression $\frac{7}{6 u^2 x} - \frac{2}{9 u x^3}$ . We'll break down the process into manageable steps, making it easier to understand and apply the concept.

The Importance of a Common Denominator

Before we can subtract the fractions, we need to find a common denominator. The common denominator is the least common multiple (LCM) of the denominators of the two fractions. In this case, the denominators are $6 u^2 x$ and $9 u x^3$ . To find the LCM, we need to factor each denominator and identify the common factors.

Factoring the Denominators

Let's factor the denominators:

$6 u^2 x = 2 \cdot 3 \cdot u^2 \cdot x$
$9 u x^3 = 3^2 \cdot u \cdot x^3$

Identifying the Common Factors

The common factors between the two denominators are $3$ , $u$ , and $x$ . We can use these common factors to find the LCM.

Finding the Least Common Multiple (LCM)

The LCM of $6 u^2 x$ and $9 u x^3$ is $2 \cdot 3^2 \cdot u^2 \cdot x^3 = 18 u^2 x^3$ .

Rewriting the Fractions with the Common Denominator

Now that we have the LCM, we can rewrite each fraction with the common denominator:

$\frac{7}{6 u^2 x} = \frac{7 \cdot 3}{6 u^2 x \cdot 3} = \frac{21}{18 u^2 x^3}$
$\frac{2}{9 u x^3} = \frac{2 \cdot 2}{9 u x^3 \cdot 2} = \frac{4}{18 u x^3}$

Subtracting the Fractions

Now that the fractions have the same denominator, we can subtract them:

$\frac{21}{18 u^2 x^3} - \frac{4}{18 u x^3} = \frac{21 - 4}{18 u x^3} = \frac{17}{18 u x^3}$

Simplifying the Result

The result is already simplified, but we can further simplify it by dividing the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD is $1$ , so the result is already in its simplest form.

Conclusion

Real-World Applications

Understanding how to subtract fractions with variables is essential in various real-world applications, such as:

Science: When working with scientific formulas, you may encounter fractions with variables that need to be subtracted.
Engineering: Engineers often use mathematical formulas to design and optimize systems, and subtracting fractions with variables is a crucial step in this process.
Finance: In finance, you may need to subtract fractions with variables when working with financial formulas or models.

Tips and Tricks

Here are some tips and tricks to help you master subtracting fractions with variables:

Practice, practice, practice: The more you practice subtracting fractions with variables, the more comfortable you'll become with the process.
Use visual aids: Visual aids, such as diagrams or charts, can help you understand the process and identify common factors.
Break down the problem: Break down the problem into smaller, manageable steps, and focus on one step at a time.

Common Mistakes to Avoid

Here are some common mistakes to avoid when subtracting fractions with variables:

Not finding the common denominator: Failing to find the common denominator can lead to incorrect results.
Not rewriting the fractions with the common denominator: Failing to rewrite the fractions with the common denominator can lead to incorrect results.
Not simplifying the result: Failing to simplify the result can lead to unnecessary complexity.

Conclusion

Subtracting fractions with variables requires finding a common denominator and rewriting each fraction with that denominator. Once the fractions have the same denominator, we can subtract them. In this article, we walked through the process of subtracting the expression $\frac{7}{6 u^2 x} - \frac{2}{9 u x^3}$ , and we arrived at the simplified result $\frac{17}{18 u x^3}$ . By following the steps outlined in this article, you'll be able to master subtracting fractions with variables and apply this skill in various real-world applications.

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Q: What is the first step in subtracting fractions with variables?

A: The first step in subtracting fractions with variables is to find the least common multiple (LCM) of the denominators. This will give you the common denominator that you need to rewrite each fraction with.

Q: How do I find the LCM of the denominators?

A: To find the LCM of the denominators, you need to factor each denominator and identify the common factors. Then, you can multiply the common factors together to get the LCM.

Q: What if the denominators have different variables?

A: If the denominators have different variables, you need to find the LCM of the variables as well as the coefficients. For example, if one denominator has $x^2$ and the other has $x^3$ , you need to find the LCM of $x^2$ and $x^3$ , which is $x^3$ .

Q: Can I simplify the result after subtracting the fractions?

A: Yes, you can simplify the result after subtracting the fractions by dividing the numerator and denominator by their greatest common divisor (GCD). This will give you the result in its simplest form.

Q: What if the result is a fraction with a variable in the denominator?

A: If the result is a fraction with a variable in the denominator, you need to simplify the fraction by canceling out any common factors between the numerator and denominator.

Q: Can I use a calculator to subtract fractions with variables?

A: Yes, you can use a calculator to subtract fractions with variables. However, it's always a good idea to check your work by hand to make sure you get the correct result.

Q: What are some common mistakes to avoid when subtracting fractions with variables?

A: Some common mistakes to avoid when subtracting fractions with variables include:

Not finding the common denominator
Not rewriting the fractions with the common denominator
Not simplifying the result
Not canceling out common factors between the numerator and denominator

Q: How can I practice subtracting fractions with variables?

A: You can practice subtracting fractions with variables by working through examples and exercises. You can also use online resources or math software to help you practice.

Q: What are some real-world applications of subtracting fractions with variables?

A: Some real-world applications of subtracting fractions with variables include:

Science: When working with scientific formulas, you may encounter fractions with variables that need to be subtracted.
Engineering: Engineers often use mathematical formulas to design and optimize systems, and subtracting fractions with variables is a crucial step in this process.
Finance: In finance, you may need to subtract fractions with variables when working with financial formulas or models.

Q: Can I use subtracting fractions with variables in algebraic expressions?

A: Yes, you can use subtracting fractions with variables in algebraic expressions. For example, you can subtract fractions with variables in a polynomial expression or in a rational expression.

Q: What are some tips for mastering subtracting fractions with variables?

A: Some tips for mastering subtracting fractions with variables include:

Practice, practice, practice: The more you practice subtracting fractions with variables, the more comfortable you'll become with the process.
Use visual aids: Visual aids, such as diagrams or charts, can help you understand the process and identify common factors.
Break down the problem: Break down the problem into smaller, manageable steps, and focus on one step at a time.

Q: Can I use subtracting fractions with variables in calculus?

A: Yes, you can use subtracting fractions with variables in calculus. For example, you can use subtracting fractions with variables when working with limits or when differentiating or integrating functions that contain fractions with variables.