Simplify The Expression Below: \left(\frac{2n}{8\pi+4}\right)\left(\frac{9r+2}{3n-2}\right ]What Is The Numerator Of The Simplified Expression?

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of various mathematical concepts, including fractions, algebraic manipulation, and factoring. In this article, we will focus on simplifying the given expression and finding the numerator of the simplified expression.

Understanding the Expression

The given expression is a product of two fractions:

$$\left(\frac{2n}{8\pi+4}\right)\left(\frac{9r+2}{3n-2}\right)$$

To simplify this expression, we need to apply the rules of algebra and fraction manipulation.

Step 1: Factor the Denominators

The first step in simplifying the expression is to factor the denominators of both fractions. The denominator of the first fraction is $8\pi+4$, which can be factored as $4(2\pi+1)$. The denominator of the second fraction is $3n-2$, which cannot be factored further.

Step 2: Simplify the Expression

Now that we have factored the denominators, we can simplify the expression by canceling out any common factors between the numerators and denominators.

$$\left(\frac{2n}{4(2\pi+1)}\right)\left(\frac{9r+2}{3n-2}\right)$$

Step 3: Cancel Out Common Factors

We can cancel out the common factor of $2$ between the numerator and denominator of the first fraction.

$$\left(\frac{n}{2(2\pi+1)}\right)\left(\frac{9r+2}{3n-2}\right)$$

Step 4: Multiply the Numerators and Denominators

Now that we have simplified the expression, we can multiply the numerators and denominators to get the final expression.

$$\frac{n(9r+2)}{2(2\pi+1)(3n-2)}$$

Finding the Numerator of the Simplified Expression

The numerator of the simplified expression is $n(9r+2)$.

Conclusion

In this article, we simplified the given expression and found the numerator of the simplified expression. We applied the rules of algebra and fraction manipulation to simplify the expression and cancel out common factors. The final expression is $\frac{n(9r+2)}{2(2\pi+1)(3n-2)}$, and the numerator of the simplified expression is $n(9r+2)$.

Final Answer

The final answer is: $\boxed{n(9r+2)}$

Introduction

In our previous article, we simplified the given expression and found the numerator of the simplified expression. However, we received many questions from readers who wanted to know more about the simplification process and the final answer. In this article, we will answer some of the most frequently asked questions about simplifying the expression.

Q: What is the final answer to the simplified expression?

A: The final answer to the simplified expression is $\frac{n(9r+2)}{2(2\pi+1)(3n-2)}$.

Q: How do I simplify the expression if the denominators are not factorable?

A: If the denominators are not factorable, you can still simplify the expression by canceling out any common factors between the numerators and denominators. However, if there are no common factors, you may need to use other techniques such as multiplying the numerator and denominator by a conjugate or using a different method to simplify the expression.

Q: Can I simplify the expression if the variables are not the same?

A: Yes, you can simplify the expression even if the variables are not the same. However, you may need to use different techniques such as multiplying the numerator and denominator by a conjugate or using a different method to simplify the expression.

Q: How do I know if the expression is simplified?

A: You can check if the expression is simplified by looking for any common factors between the numerators and denominators. If there are no common factors, you can try multiplying the numerator and denominator by a conjugate or using a different method to simplify the expression.

Q: Can I use a calculator to simplify the expression?

A: Yes, you can use a calculator to simplify the expression. However, keep in mind that calculators may not always give you the simplest form of the expression. It's always a good idea to check your work by hand to make sure the expression is simplified correctly.

Q: How do I apply the rules of algebra and fraction manipulation to simplify the expression?

A: To apply the rules of algebra and fraction manipulation, you need to follow these steps:

1. Factor the denominators of both fractions.
2. Simplify the expression by canceling out any common factors between the numerators and denominators.
3. Multiply the numerators and denominators to get the final expression.

Q: What are some common mistakes to avoid when simplifying the expression?

A: Some common mistakes to avoid when simplifying the expression include:

• Not factoring the denominators correctly
• Not canceling out common factors between the numerators and denominators
• Not multiplying the numerator and denominator by a conjugate when necessary
• Not checking the work by hand to make sure the expression is simplified correctly

Conclusion

In this article, we answered some of the most frequently asked questions about simplifying the expression. We provided tips and techniques for simplifying the expression, including factoring the denominators, canceling out common factors, and multiplying the numerator and denominator by a conjugate. We also discussed common mistakes to avoid when simplifying the expression.

Final Answer

The final answer is: $\boxed{\frac{n(9r+2)}{2(2\pi+1)(3n-2)}}$