Subtract And Simplify The Following Expression: 7 9 U X 3 − 2 3 U 2 X \frac{7}{9ux^3} - \frac{2}{3u^2x} 9 U X 3 7 − 3 U 2 X 2

Mar 11, 2025 by ADMIN 130 views

**Subtract and Simplify the Given Algebraic Expression**

Understanding the Problem

When dealing with algebraic expressions, it's essential to understand the rules of arithmetic operations, including addition, subtraction, multiplication, and division. In this article, we will focus on subtracting and simplifying a given algebraic expression involving fractions. The expression to be simplified is $\frac{7}{9ux^3} - \frac{2}{3u^2x}$ .

The Importance of Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial step in solving mathematical problems. It helps to make the expressions more manageable and easier to work with. By simplifying expressions, we can identify common factors, cancel out terms, and make the problem more tractable. In this case, we need to subtract the two fractions and simplify the resulting expression.

Subtracting Fractions with Different Denominators

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

Finding the Least Common Multiple (LCM)

To find the LCM of $9ux^3$ and $3u^2x$ , we need to factorize the denominators:

$9ux^3 = 3 \times 3 \times u \times x \times x \times x$
$3u^2x = 3 \times u \times u \times x$

The common factors are $3$ , $u$ , and $x$ . Therefore, the LCM is $3 \times 3 \times u \times u \times x \times x \times x = 9u^2x^3$ .

Subtracting the Fractions

Now that we have the common denominator, we can subtract the fractions:

$\frac{7}{9ux^3} - \frac{2}{3u^2x} = \frac{7}{9ux^3} - \frac{2 \times 3x^2}{3u^2x \times 3x^2}$

Simplifying the second fraction, we get:

$\frac{7}{9ux^3} - \frac{6x^2}{9u^2x^3}$

Combining the Fractions

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

$\frac{7}{9ux^3} - \frac{6x^2}{9u^2x^3} = \frac{7 - 6x^2}{9ux^3}$

Simplifying the Expression

The expression is now simplified, and we can see that the numerator is $7 - 6x^2$ . However, we can further simplify the expression by factoring out the common factor $7$ from the numerator:

$\frac{7 - 6x^2}{9ux^3} = \frac{7(1 - \frac{6}{7}x^2)}{9ux^3}$

Final Simplified Expression

The final simplified expression is $\frac{7(1 - \frac{6}{7}x^2)}{9ux^3}$ .

Conclusion

In this article, we subtracted and simplified the given algebraic expression $\frac{7}{9ux^3} - \frac{2}{3u^2x}$ . We found the least common multiple (LCM) of the denominators, subtracted the fractions, and simplified the resulting expression. The final simplified expression is $\frac{7(1 - \frac{6}{7}x^2)}{9ux^3}$ . This expression is now in its simplest form, and we can use it to solve mathematical problems involving algebraic expressions.

Future Applications

The skills learned in this article can be applied to various mathematical problems involving algebraic expressions. By understanding the rules of arithmetic operations and simplifying expressions, we can solve complex mathematical problems and make the problem more tractable. The ability to simplify algebraic expressions is a crucial skill in mathematics, and it's essential to practice and master this skill to become proficient in mathematics.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's essential to avoid common mistakes. Some common mistakes include:

Not finding the least common multiple (LCM) of the denominators
Not simplifying the numerator and denominator separately
Not factoring out common factors from the numerator and denominator

By avoiding these common mistakes, we can ensure that our simplified expressions are accurate and correct.

Real-World Applications

The skills learned in this article have real-world applications in various fields, including:

Physics: Algebraic expressions are used to describe the motion of objects and the behavior of physical systems.
Engineering: Algebraic expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Algebraic expressions are used to model economic systems and make predictions about economic trends.

By understanding and simplifying algebraic expressions, we can solve complex mathematical problems and make the problem more tractable. The skills learned in this article can be applied to various mathematical problems and have real-world applications in various fields.

Q: What is the first step in subtracting and simplifying algebraic expressions?

A: The first step in subtracting and simplifying algebraic expressions is to find the least common multiple (LCM) of the denominators. This will allow us to combine the fractions and simplify the expression.

Q: How do I find the least common multiple (LCM) of the denominators?

A: To find the LCM of the denominators, we need to factorize the denominators and identify the common factors. We can then multiply the common factors together to find the LCM.

Q: What is the difference between a common denominator and a least common multiple (LCM)?

A: A common denominator is a denominator that is shared by two or more fractions, while a least common multiple (LCM) is the smallest multiple that is shared by two or more numbers. In the context of algebraic expressions, the LCM is used to find a common denominator.

Q: How do I simplify the numerator and denominator separately?

A: To simplify the numerator and denominator separately, we need to factor out common factors from the numerator and denominator. We can then cancel out any common factors to simplify the expression.

Q: What is the final simplified expression for the given algebraic expression $\frac{7}{9ux^3} - \frac{2}{3u^2x}$ ?

A: The final simplified expression for the given algebraic expression $\frac{7}{9ux^3} - \frac{2}{3u^2x}$ is $\frac{7(1 - \frac{6}{7}x^2)}{9ux^3}$ .

Q: How do I apply the skills learned in this article to real-world problems?

A: The skills learned in this article can be applied to various mathematical problems involving algebraic expressions. By understanding the rules of arithmetic operations and simplifying expressions, we can solve complex mathematical problems and make the problem more tractable. The ability to simplify algebraic expressions is a crucial skill in mathematics, and it's essential to practice and master this skill to become proficient in mathematics.

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

Not finding the least common multiple (LCM) of the denominators
Not simplifying the numerator and denominator separately
Not factoring out common factors from the numerator and denominator

Q: How do I ensure that my simplified expressions are accurate and correct?

A: To ensure that your simplified expressions are accurate and correct, you should:

Double-check your work to ensure that you have found the correct least common multiple (LCM) of the denominators
Simplify the numerator and denominator separately to ensure that you have factored out all common factors
Check your work to ensure that you have not made any errors in simplifying the expression

Q: What are some real-world applications of the skills learned in this article?

A: The skills learned in this article have real-world applications in various fields, including:

Physics: Algebraic expressions are used to describe the motion of objects and the behavior of physical systems.
Engineering: Algebraic expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
Economics: Algebraic expressions are used to model economic systems and make predictions about economic trends.