What Is The Product Of The Rational Expressions Shown Below? Make Sure Your Answer Is In Reduced Form. P − 3 R + 7 ⋅ 2 R X − 1 \frac{p-3}{r+7} \cdot \frac{2r}{x-1} R + 7 P − 3 ​ ⋅ X − 1 2 R ​ A. 2 X + 7 \frac{2}{x+7} X + 7 2 ​ B. 2 2 = 3 \frac{2}{2=3} 2 = 3 2 ​ C. 2 R Π − 1 \frac{2r}{\pi-1} Π − 1 2 R ​ 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 a fundamental concept in algebra. When we multiply rational expressions, we need to follow specific rules to simplify the resulting expression.

Multiplying Rational Expressions

To multiply rational expressions, we need to multiply the numerators together and the denominators together. This is based on the rule that when we multiply fractions, we multiply the numerators and denominators separately. The resulting expression may or may not be in its simplest form.

The Given Rational Expressions

We are given two rational expressions:

$\frac{p-3}{r+7}$ and $\frac{2r}{x-1}$

We need to find the product of these two expressions.

Multiplying the Numerators and Denominators

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

$(p-3) \cdot 2r$ and $(r+7) \cdot (x-1)$

Simplifying the Expression

Now, we simplify the resulting expression:

$2r(p-3)$ and $(r+7)(x-1)$

Distributing the Terms

To simplify further, we distribute the terms in the numerators and denominators:

$2rp - 6r$ and $rx - r - 7x + 7$

Combining Like Terms

Now, we combine like terms in the numerator and denominator:

$2rp - 6r$ and $rx - 7x - r + 7$

Writing the Final Expression

The final expression is:

$\frac{2rp - 6r}{rx - 7x - r + 7}$

Reducing the Expression

To reduce the expression, we need to find the greatest common factor (GCF) of the numerator and denominator. The GCF is the largest factor that divides both the numerator and denominator.

Finding the GCF

The GCF of $2rp - 6r$ and $rx - 7x - r + 7$ is $r$.

Dividing the Numerator and Denominator by the GCF

Now, we divide the numerator and denominator by the GCF:

$\frac{2p - 6}{x - 7 - 1}$

Simplifying the Expression

Now, we simplify the expression:

$\frac{2p - 6}{x - 8}$

The Final Answer

The final answer is $\frac{2(p-3)}{x-8}$.

Comparison with the Options

Now, let's compare the final answer with the options:

A. $\frac{2}{x+7}$

B. $\frac{2}{2=3}$

C. $\frac{2r}{\pi-1}$

D. None of the above

Conclusion

The final answer is $\frac{2(p-3)}{x-8}$, which is not among the options. Therefore, the correct answer is D. None of the above.

Final Thoughts

Multiplying rational expressions requires following specific rules to simplify the resulting expression. In this problem, we multiplied two rational expressions and simplified the resulting expression. The final answer is $\frac{2(p-3)}{x-8}$, which is not among the options. Therefore, the correct answer is D. None of the above.

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 when we multiply fractions, we multiply the numerators and denominators separately.

Q: How do I simplify the resulting expression after multiplying rational expressions?

A: To simplify the resulting expression, you need to combine like terms in the numerator and denominator. You also need to find the greatest common factor (GCF) of the numerator and denominator and divide both by the GCF.

Q: What is the greatest common factor (GCF) and why is it important?

A: The GCF is the largest factor that divides both the numerator and denominator. It is important because dividing both the numerator and denominator by the GCF simplifies the expression and makes it easier to work with.

Q: How do I find the GCF of two expressions?

A: To find the GCF of two expressions, you need to list all the factors of each expression and find the largest factor that is common to both.

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 combining the numerators.

Q: Can I multiply rational expressions with different variables?

A: Yes, you can multiply rational expressions with different variables. However, you need to be careful when simplifying the resulting expression, as the variables may not cancel out.

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

A: A rational expression is in its simplest form if the numerator and denominator have no common factors other than 1.

Q: Can I multiply rational expressions with negative exponents?

A: Yes, you can multiply rational expressions with negative exponents. However, you need to be careful when simplifying the resulting expression, as the negative exponents may change the sign of the expression.

Q: How do I multiply rational expressions with fractions in the numerator or denominator?

A: To multiply rational expressions with fractions in the numerator or denominator, you need to multiply the fractions together and simplify the resulting expression.

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

A: Yes, you can multiply rational expressions with decimals in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the decimals may change the value of the expression.

Q: How do I multiply rational expressions with variables in the numerator or denominator?

A: To multiply rational expressions with variables in the numerator or denominator, you need to multiply the variables together and simplify the resulting expression.

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

A: Yes, you can multiply rational expressions with complex numbers in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the complex numbers may change the value of the expression.

Q: How do I multiply rational expressions with exponents in the numerator or denominator?

A: To multiply rational expressions with exponents in the numerator or denominator, you need to multiply the exponents together and simplify the resulting expression.

Q: Can I multiply rational expressions with absolute values in the numerator or denominator?

A: Yes, you can multiply rational expressions with absolute values in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the absolute values may change the value of the expression.

Q: How do I multiply rational expressions with square roots in the numerator or denominator?

A: To multiply rational expressions with square roots in the numerator or denominator, you need to multiply the square roots together and simplify the resulting expression.

Q: Can I multiply rational expressions with pi in the numerator or denominator?

A: Yes, you can multiply rational expressions with pi in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the pi may change the value of the expression.

Q: How do I multiply rational expressions with e in the numerator or denominator?

A: To multiply rational expressions with e in the numerator or denominator, you need to multiply the e together and simplify the resulting expression.

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

A: Yes, you can multiply rational expressions with trigonometric functions in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the trigonometric functions may change the value of the expression.

Q: How do I multiply rational expressions with logarithmic functions in the numerator or denominator?

A: To multiply rational expressions with logarithmic functions in the numerator or denominator, you need to multiply the logarithmic functions together and simplify the resulting expression.

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

A: Yes, you can multiply rational expressions with inverse functions in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the inverse functions may change the value of the expression.

Q: How do I multiply rational expressions with composite functions in the numerator or denominator?

A: To multiply rational expressions with composite functions in the numerator or denominator, you need to multiply the composite functions together and simplify the resulting expression.

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

A: Yes, you can multiply rational expressions with parametric functions in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the parametric functions may change the value of the expression.

Q: How do I multiply rational expressions with implicit functions in the numerator or denominator?

A: To multiply rational expressions with implicit functions in the numerator or denominator, you need to multiply the implicit functions together and simplify the resulting expression.

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

A: Yes, you can multiply rational expressions with explicit functions in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the explicit functions may change the value of the expression.

Q: How do I multiply rational expressions with piecewise functions in the numerator or denominator?

A: To multiply rational expressions with piecewise functions in the numerator or denominator, you need to multiply the piecewise functions together and simplify the resulting expression.

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

A: Yes, you can multiply rational expressions with vector functions in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the vector functions may change the value of the expression.

Q: How do I multiply rational expressions with matrix functions in the numerator or denominator?

A: To multiply rational expressions with matrix functions in the numerator or denominator, you need to multiply the matrix functions together and simplify the resulting expression.

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

A: Yes, you can multiply rational expressions with differential functions in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the differential functions may change the value of the expression.

Q: How do I multiply rational expressions with integral functions in the numerator or denominator?

A: To multiply rational expressions with integral functions in the numerator or denominator, you need to multiply the integral functions together and simplify the resulting expression.

Q: Can I multiply rational expressions with differential equations in the numerator or denominator?

A: Yes, you can multiply rational expressions with differential equations in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the differential equations may change the value of the expression.

Q: How do I multiply rational expressions with integral equations in the numerator or denominator?

A: To multiply rational expressions with integral equations in the numerator or denominator, you need to multiply the integral equations together and simplify the resulting expression.

Q: Can I multiply rational expressions with parametric equations in the numerator or denominator?

A: Yes, you can multiply rational expressions with parametric equations in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the parametric equations may change the value of the expression.

Q: How do I multiply rational expressions with implicit equations in the numerator or denominator?

A: To multiply rational expressions with implicit equations in the numerator or denominator, you need to multiply the implicit equations together and simplify the resulting expression.

Q: Can I multiply rational expressions with explicit equations in the numerator or denominator?

A: Yes, you can multiply rational expressions with explicit equations in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the explicit equations may change the value of the expression.

Q: How do I multiply rational expressions with piecewise equations in the numerator or denominator?

A: To multiply rational expressions with piecewise equations in the numerator or denominator, you need to multiply the piecewise equations together and simplify the resulting expression.

Q: Can I multiply rational expressions with vector equations in the numerator or denominator?

A: Yes, you can multiply rational expressions with vector equations in the numerator or denominator. However, you need to be careful when simplifying the resulting expression, as the vector equations may change the value of the expression.

Q: How do I multiply rational expressions with matrix equations in the numerator or denominator?

A: To multiply rational expressions with matrix equations in the numerator or denominator, you need to multiply the matrix equations together and simplify the resulting expression.

Q: Can I multiply rational expressions with differential equations in the numerator or denominator?

A: Yes, you can multiply rational expressions with differential equations in the numerator or denominator. However, you need to be careful when simplifying the resulting