What Is The Product Of The Rational Expressions Shown Below? Make Sure Your Answer Is In Reduced Form.$\[ \frac{x-3}{x+7} \cdot \frac{2x}{x-3} \\]A. \[$\frac{2x}{x-3}\$\]B. \[$\frac{2}{x+7}\$\]C. \[$\frac{2x}{x+7}\$\]D.

Understanding Rational Expressions

Rational expressions are fractions that contain variables and constants in the numerator and denominator. They are used to represent mathematical relationships and are essential in algebra and calculus. When we multiply rational expressions, we need to follow specific rules to simplify the result.

Multiplying Rational Expressions

To multiply rational expressions, we multiply the numerators together and the denominators together. This is based on the rule that a/b * c/d = (ac)/(bd). However, we need to be careful when multiplying rational expressions, as we may end up with a fraction that is not in its simplest form.

The Product of the Given Rational Expressions

We are given two rational expressions:

$\frac{x-3}{x+7} \cdot \frac{2x}{x-3}$

To find the product of these expressions, we multiply the numerators together and the denominators together:

$\frac{(x-3) \cdot 2x}{(x+7) \cdot (x-3)}$

Simplifying the Product

Now, we need to simplify the product by canceling out any common factors in the numerator and denominator. In this case, we can see that the (x-3) terms in the numerator and denominator can be canceled out:

$\frac{(x-3) \cdot 2x}{(x+7) \cdot (x-3)} = \frac{2x}{x+7}$

Conclusion

The product of the given rational expressions is $\frac{2x}{x+7}$. This is the reduced form of the product, and it cannot be simplified further.

Final Answer

The final answer is $\boxed{\frac{2x}{x+7}}$.

Discussion

This problem requires the application of the rules for multiplying rational expressions. It also requires the ability to simplify fractions by canceling out common factors. The correct answer is $\frac{2x}{x+7}$, which is the reduced form of the product.

Related Topics

• Simplifying rational expressions
• Canceling out common factors
• Multiplying rational expressions

Example Problems

• Multiply the rational expressions $\frac{x+2}{x-3} \cdot \frac{2x}{x+2}$.
• Simplify the rational expression $\frac{2x}{x+7} \cdot \frac{x-3}{x-3}$.

Solutions

• The product of the rational expressions is $\frac{2x}{x-3}$.
• The simplified rational expression is $\frac{2x}{x+7}$.

Conclusion

In conclusion, the product of the given rational expressions is $\frac{2x}{x+7}$. This is the reduced form of the product, and it cannot be simplified further. The correct answer is $\boxed{\frac{2x}{x+7}}$.

Q: What is the rule for multiplying rational expressions?

A: The rule for multiplying rational expressions is to multiply the numerators together and the denominators together. This is based on the rule that a/b * c/d = (ac)/(bd).

Q: How do I simplify the product of rational expressions?

A: To simplify the product of rational expressions, you need to cancel out any common factors in the numerator and denominator. This involves identifying any common factors and dividing them out.

Q: What is the difference between multiplying rational expressions and adding or subtracting rational expressions?

A: Multiplying rational expressions involves multiplying the numerators together and the denominators together, whereas adding or subtracting rational expressions involves finding a common denominator and then adding or subtracting the numerators.

Q: Can I simplify a rational expression by canceling out a common factor in the numerator and denominator?

A: Yes, you can simplify a rational expression by canceling out a common factor in the numerator and denominator. This is a key step in simplifying rational expressions.

Q: How do I know if a rational expression is in its simplest form?

A: A rational expression is in its simplest form if there are no common factors in the numerator and denominator that can be canceled out.

Q: Can I multiply rational expressions with different signs in the numerator and denominator?

A: Yes, you can multiply rational expressions with different signs in the numerator and denominator. However, you need to follow the rules for multiplying negative numbers.

Q: How do I handle zero in the denominator when multiplying rational expressions?

A: When multiplying rational expressions, you need to be careful not to introduce zero in the denominator. If the denominator of one of the expressions is zero, you need to simplify the expression before multiplying.

Q: Can I multiply rational expressions with variables in the numerator and denominator?

A: Yes, you can multiply rational expressions with variables in the numerator and denominator. However, you need to follow the rules for multiplying variables.

Q: How do I simplify a rational expression with a variable in the denominator?

A: To simplify a rational expression with a variable in the denominator, you need to find a value for the variable that makes the denominator zero. This will help you to simplify the expression.

Q: Can I multiply rational expressions with fractions in the numerator and denominator?

A: Yes, you can multiply rational expressions with fractions in the numerator and denominator. However, you need to follow the rules for multiplying fractions.

Q: How do I simplify a rational expression with a fraction in the numerator and denominator?

A: To simplify a rational expression with a fraction in the numerator and denominator, you need to multiply the fractions together and then simplify the result.

Q: Can I multiply rational expressions with decimals in the numerator and denominator?

A: Yes, you can multiply rational expressions with decimals in the numerator and denominator. However, you need to follow the rules for multiplying decimals.

Q: How do I simplify a rational expression with a decimal in the numerator and denominator?

A: To simplify a rational expression with a decimal in the numerator and denominator, you need to multiply the decimals together and then simplify the result.

Q: Can I multiply rational expressions with percentages in the numerator and denominator?

A: Yes, you can multiply rational expressions with percentages in the numerator and denominator. However, you need to follow the rules for multiplying percentages.

Q: How do I simplify a rational expression with a percentage in the numerator and denominator?

A: To simplify a rational expression with a percentage in the numerator and denominator, you need to multiply the percentages together and then simplify the result.

Q: Can I multiply rational expressions with mixed numbers in the numerator and denominator?

A: Yes, you can multiply rational expressions with mixed numbers in the numerator and denominator. However, you need to follow the rules for multiplying mixed numbers.

Q: How do I simplify a rational expression with a mixed number in the numerator and denominator?

A: To simplify a rational expression with a mixed number in the numerator and denominator, you need to multiply the mixed numbers together and then simplify the result.

Q: Can I multiply rational expressions with improper fractions in the numerator and denominator?

A: Yes, you can multiply rational expressions with improper fractions in the numerator and denominator. However, you need to follow the rules for multiplying improper fractions.

Q: How do I simplify a rational expression with an improper fraction in the numerator and denominator?

A: To simplify a rational expression with an improper fraction in the numerator and denominator, you need to multiply the improper fractions together and then simplify the result.

Q: Can I multiply rational expressions with complex numbers in the numerator and denominator?

A: Yes, you can multiply rational expressions with complex numbers in the numerator and denominator. However, you need to follow the rules for multiplying complex numbers.

Q: How do I simplify a rational expression with a complex number in the numerator and denominator?

A: To simplify a rational expression with a complex number in the numerator and denominator, you need to multiply the complex numbers together and then simplify the result.

Q: Can I multiply rational expressions with matrices in the numerator and denominator?

A: Yes, you can multiply rational expressions with matrices in the numerator and denominator. However, you need to follow the rules for multiplying matrices.

Q: How do I simplify a rational expression with a matrix in the numerator and denominator?

A: To simplify a rational expression with a matrix in the numerator and denominator, you need to multiply the matrices together and then simplify the result.

Q: Can I multiply rational expressions with vectors in the numerator and denominator?

A: Yes, you can multiply rational expressions with vectors in the numerator and denominator. However, you need to follow the rules for multiplying vectors.

Q: How do I simplify a rational expression with a vector in the numerator and denominator?

A: To simplify a rational expression with a vector in the numerator and denominator, you need to multiply the vectors together and then simplify the result.

Q: Can I multiply rational expressions with polynomials in the numerator and denominator?

A: Yes, you can multiply rational expressions with polynomials in the numerator and denominator. However, you need to follow the rules for multiplying polynomials.

Q: How do I simplify a rational expression with a polynomial in the numerator and denominator?

A: To simplify a rational expression with a polynomial in the numerator and denominator, you need to multiply the polynomials together and then simplify the result.

Q: Can I multiply rational expressions with functions in the numerator and denominator?

A: Yes, you can multiply rational expressions with functions in the numerator and denominator. However, you need to follow the rules for multiplying functions.

Q: How do I simplify a rational expression with a function in the numerator and denominator?

A: To simplify a rational expression with a function in the numerator and denominator, you need to multiply the functions together and then simplify the result.

Q: Can I multiply rational expressions with trigonometric functions in the numerator and denominator?

A: Yes, you can multiply rational expressions with trigonometric functions in the numerator and denominator. However, you need to follow the rules for multiplying trigonometric functions.

Q: How do I simplify a rational expression with a trigonometric function in the numerator and denominator?

A: To simplify a rational expression with a trigonometric function in the numerator and denominator, you need to multiply the trigonometric functions together and then simplify the result.

Q: Can I multiply rational expressions with exponential functions in the numerator and denominator?

A: Yes, you can multiply rational expressions with exponential functions in the numerator and denominator. However, you need to follow the rules for multiplying exponential functions.

Q: How do I simplify a rational expression with an exponential function in the numerator and denominator?

A: To simplify a rational expression with an exponential function in the numerator and denominator, you need to multiply the exponential functions together and then simplify the result.

Q: Can I multiply rational expressions with logarithmic functions in the numerator and denominator?

A: Yes, you can multiply rational expressions with logarithmic functions in the numerator and denominator. However, you need to follow the rules for multiplying logarithmic functions.

Q: How do I simplify a rational expression with a logarithmic function in the numerator and denominator?

A: To simplify a rational expression with a logarithmic function in the numerator and denominator, you need to multiply the logarithmic functions together and then simplify the result.

Q: Can I multiply rational expressions with inverse functions in the numerator and denominator?

A: Yes, you can multiply rational expressions with inverse functions in the numerator and denominator. However, you need to follow the rules for multiplying inverse functions.

Q: How do I simplify a rational expression with an inverse function in the numerator and denominator?

A: To simplify a rational expression with an inverse function in the numerator and denominator, you need to multiply the inverse functions together and then simplify the result.

Q: Can I multiply rational expressions with composite functions in the numerator and denominator?

A: Yes, you can multiply rational expressions with composite functions in the numerator and denominator. However, you need to follow the rules for multiplying composite functions.

Q: How do I simplify a rational expression with a composite function in the numerator and denominator?

A: To simplify a rational expression with a composite function in the numerator and denominator, you need to multiply the composite functions together and then simplify the result.

Q: Can I multiply rational expressions with parametric functions in the numerator and denominator?

A: Yes, you can multiply rational expressions with parametric functions in the numerator and denominator. However, you need to follow the rules for multiplying parametric functions.

Q: How do I simplify a rational expression with a parametric function in the numerator and denominator?

A: To simplify a rational expression with a parametric function in the numerator and denominator, you need to multiply the parametric functions together and then simplify the result