If \[$\bar{B}\$\] And \[$\bar{D}\$\] Are Rational Expressions, Then:$\[\frac{A}{B} \div \frac{C}{D} = \frac{A \cdot D}{B \cdot C}\\]A. True B. False

Feb 28, 2025 by ADMIN 150 views

**Rational Expression Division: Understanding the Rules**

Introduction

In mathematics, rational expressions are a crucial part of algebraic manipulations. When dealing with rational expressions, it's essential to understand the rules governing their operations, including division. In this article, we will explore the division of rational expressions and determine whether the given statement is true or false.

What are Rational Expressions?

Rational expressions are fractions that contain variables or constants in the numerator and/or denominator. They can be expressed in the form of $\frac{A}{B}$ , where $A$ and $B$ are polynomials. Rational expressions can be simplified, added, subtracted, multiplied, and divided, just like regular fractions.

Division of Rational Expressions

When dividing rational expressions, we need to follow a specific rule. The rule states that to divide two rational expressions, we multiply the first expression by the reciprocal of the second expression. In other words, $\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}$ .

The Given Statement

The given statement is: $\frac{A}{B} \div \frac{C}{D} = \frac{A \cdot D}{B \cdot C}$ . We need to determine whether this statement is true or false.

Analysis

Let's analyze the given statement. When we divide two rational expressions, we multiply the first expression by the reciprocal of the second expression. This means that $\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}$ .

Now, let's simplify the expression $\frac{A}{B} \cdot \frac{D}{C}$ . We can do this by multiplying the numerators and denominators separately. This gives us $\frac{A \cdot D}{B \cdot C}$ .

Conclusion

Based on our analysis, we can see that the given statement is actually true. When we divide two rational expressions, we can simplify the expression by multiplying the first expression by the reciprocal of the second expression, which results in $\frac{A \cdot D}{B \cdot C}$ .

Example

Let's consider an example to illustrate this concept. Suppose we want to divide the rational expressions $\frac{x+2}{x-1}$ and $\frac{x+3}{x+2}$ . We can do this by multiplying the first expression by the reciprocal of the second expression:

$\frac{x+2}{x-1} \div \frac{x+3}{x+2} = \frac{x+2}{x-1} \cdot \frac{x+2}{x+3}$

Simplifying this expression, we get:

$\frac{(x+2)^2}{(x-1)(x+3)}$

This is the result of dividing the two rational expressions.

Conclusion

In conclusion, the given statement is true. When we divide two rational expressions, we can simplify the expression by multiplying the first expression by the reciprocal of the second expression, which results in $\frac{A \cdot D}{B \cdot C}$ . This rule is essential in algebraic manipulations and is used extensively in mathematics.

Final Answer

Q&A: Rational Expression Division

Q: What is the rule for dividing rational expressions?

A: The rule for dividing rational expressions states that to divide two rational expressions, we multiply the first expression by the reciprocal of the second expression. In other words, $\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \cdot \frac{D}{C}$ .

Q: How do I simplify a rational expression after dividing?

A: To simplify a rational expression after dividing, we can multiply the numerators and denominators separately. This gives us $\frac{A \cdot D}{B \cdot C}$ .

Q: What if the denominators of the two rational expressions are the same?

A: If the denominators of the two rational expressions are the same, we can simply divide the numerators. In other words, $\frac{A}{B} \div \frac{C}{B} = \frac{A}{C}$ .

Q: Can I divide a rational expression by a polynomial?

A: Yes, you can divide a rational expression by a polynomial. However, you need to follow the same rule as before: multiply the rational expression by the reciprocal of the polynomial.

Q: What if I have a rational expression with a variable in the denominator?

A: If you have a rational expression with a variable in the denominator, you need to be careful when dividing. You may need to multiply both the numerator and denominator by the conjugate of the denominator to eliminate the variable in the denominator.

Q: Can I divide a rational expression with a negative exponent?

A: Yes, you can divide a rational expression with a negative exponent. When dividing, you need to follow the rule that $a^{-n} = \frac{1}{a^n}$ .

Q: What if I have a rational expression with a fraction in the denominator?

A: If you have a rational expression with a fraction in the denominator, you need to follow the same rule as before: multiply the rational expression by the reciprocal of the fraction in the denominator.

Q: Can I divide a rational expression with a complex number in the denominator?

A: Yes, you can divide a rational expression with a complex number in the denominator. However, you need to follow the same rule as before: multiply the rational expression by the reciprocal of the complex number in the denominator.

Conclusion

In conclusion, dividing rational expressions is a crucial part of algebraic manipulations. By following the rule for dividing rational expressions and simplifying the expression after dividing, you can solve a wide range of problems involving rational expressions.

Final Tips

Always follow the rule for dividing rational expressions.
Simplify the expression after dividing by multiplying the numerators and denominators separately.
Be careful when dividing rational expressions with variables or complex numbers in the denominator.
Practice, practice, practice! Dividing rational expressions is a skill that requires practice to develop.

Common Mistakes

Forgetting to multiply the rational expression by the reciprocal of the second expression.
Not simplifying the expression after dividing.
Not being careful when dividing rational expressions with variables or complex numbers in the denominator.