Simplify The Expression Below: \left(\frac{2n}{6n+4}\right)\left(\frac{3n+2}{3n-2}\right ]What Is The Numerator Of The Simplified Expression?\begin{tabular}{|l|}\hline 2 \$n$ \$3n-2$ \$3n+2$

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Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. In this article, we will focus on simplifying a given expression and finding the numerator of the simplified expression. We will break down the problem into manageable steps, and by the end of this article, you will have a clear understanding of how to simplify complex expressions.

The Given Expression

The given expression is:

(2n6n+4)(3n+23n−2)\left(\frac{2n}{6n+4}\right)\left(\frac{3n+2}{3n-2}\right)

Our goal is to simplify this expression and find the numerator of the simplified expression.

Step 1: Factor the Numerator and Denominator

To simplify the expression, we need to factor the numerator and denominator of each fraction. Let's start with the first fraction:

2n6n+4\frac{2n}{6n+4}

We can factor out the greatest common factor (GCF) of the numerator and denominator, which is 2n:

2n6n+4=2n2(3n+2)\frac{2n}{6n+4} = \frac{2n}{2(3n+2)}

Now, let's factor the second fraction:

3n+23n−2\frac{3n+2}{3n-2}

We can factor out the GCF of the numerator and denominator, which is (3n-2):

3n+23n−2=(3n−2)+43n−2\frac{3n+2}{3n-2} = \frac{(3n-2)+4}{3n-2}

Step 2: Cancel Out Common Factors

Now that we have factored the numerator and denominator of each fraction, we can cancel out common factors. Let's start with the first fraction:

2n2(3n+2)\frac{2n}{2(3n+2)}

We can cancel out the common factor of 2:

2n2(3n+2)=n3n+2\frac{2n}{2(3n+2)} = \frac{n}{3n+2}

Now, let's look at the second fraction:

(3n−2)+43n−2\frac{(3n-2)+4}{3n-2}

We can cancel out the common factor of (3n-2):

(3n−2)+43n−2=43n−2\frac{(3n-2)+4}{3n-2} = \frac{4}{3n-2}

Step 3: Multiply the Fractions

Now that we have simplified each fraction, we can multiply them together:

n3n+2⋅43n−2\frac{n}{3n+2} \cdot \frac{4}{3n-2}

To multiply fractions, we need to multiply the numerators and denominators separately:

n⋅4(3n+2)⋅(3n−2)\frac{n \cdot 4}{(3n+2) \cdot (3n-2)}

Step 4: Simplify the Expression

Now that we have multiplied the fractions, we can simplify the expression by canceling out common factors. Let's look at the numerator:

nâ‹…4=4nn \cdot 4 = 4n

And let's look at the denominator:

(3n+2)⋅(3n−2)=9n2−4(3n+2) \cdot (3n-2) = 9n^2 - 4

We can see that there are no common factors between the numerator and denominator, so the expression is already simplified.

Conclusion

The simplified expression is:

4n9n2−4\frac{4n}{9n^2 - 4}

The numerator of the simplified expression is 4n.

Final Answer

The final answer is: 4n\boxed{4n}

Introduction

In our previous article, we simplified the given expression and found the numerator of the simplified expression. However, we understand that sometimes, it's not enough to just provide the final answer. You may have questions about the steps involved in simplifying the expression, or you may want to know more about the underlying concepts. In this article, we will address some of the most frequently asked questions about simplifying the expression.

Q&A

Q: What is the greatest common factor (GCF) of the numerator and denominator of each fraction?

A: The GCF of the numerator and denominator of each fraction is the largest number that divides both the numerator and denominator without leaving a remainder. In the case of the first fraction, the GCF is 2n, and in the case of the second fraction, the GCF is (3n-2).

Q: Why do we need to factor the numerator and denominator of each fraction?

A: Factoring the numerator and denominator of each fraction allows us to cancel out common factors, which simplifies the expression. By factoring, we can identify the common factors and cancel them out, making the expression easier to work with.

Q: What is the difference between canceling out common factors and simplifying an expression?

A: Canceling out common factors is a step in simplifying an expression. When we cancel out common factors, we are essentially removing the common factors from the numerator and denominator, which simplifies the expression. However, simplifying an expression also involves combining like terms and eliminating any remaining common factors.

Q: How do we know when to cancel out common factors?

A: We know when to cancel out common factors when we have factored the numerator and denominator of each fraction and have identified the common factors. We can then cancel out the common factors by dividing both the numerator and denominator by the common factor.

Q: What is the final answer to the problem?

A: The final answer to the problem is 4n\boxed{4n}, which is the numerator of the simplified expression.

Q: Can you provide more examples of simplifying expressions?

A: Yes, we can provide more examples of simplifying expressions. Here are a few examples:

  • (2x4x+2)(3x+13x−1)\left(\frac{2x}{4x+2}\right)\left(\frac{3x+1}{3x-1}\right)
  • (3y6y+3)(2y+12y−1)\left(\frac{3y}{6y+3}\right)\left(\frac{2y+1}{2y-1}\right)
  • (4z8z+4)(3z+13z−1)\left(\frac{4z}{8z+4}\right)\left(\frac{3z+1}{3z-1}\right)

We can simplify each of these expressions by following the same steps as before: factoring the numerator and denominator of each fraction, canceling out common factors, and multiplying the fractions together.

Conclusion

Simplifying expressions is an important skill in mathematics, and it requires a deep understanding of the underlying concepts. By following the steps outlined in this article, you can simplify complex expressions and find the numerator of the simplified expression. Remember to factor the numerator and denominator of each fraction, cancel out common factors, and multiply the fractions together. With practice and patience, you will become proficient in simplifying expressions and solving complex problems.

Final Answer

The final answer is: 4n\boxed{4n}