What Is The Product? 4 N 4 N − 4 ⋅ N − 1 N + 1 \frac{4n}{4n-4} \cdot \frac{n-1}{n+1} 4 N − 4 4 N ​ ⋅ N + 1 N − 1 ​ A. 4 N N + 1 \frac{4n}{n+1} N + 1 4 N ​ B. N N + 1 \frac{n}{n+1} N + 1 N ​ C. 1 N + 1 \frac{1}{n+1} N + 1 1 ​ D. 4 N + 1 \frac{4}{n+1} N + 1 4 ​

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for students and professionals alike. In this article, we will focus on simplifying a specific algebraic expression, which is given as 4n4n−4⋅n−1n+1\frac{4n}{4n-4} \cdot \frac{n-1}{n+1}. We will break down the steps involved in simplifying this expression and provide a clear explanation of each step.

Understanding the Expression

The given expression is a product of two fractions, 4n4n−4\frac{4n}{4n-4} and n−1n+1\frac{n-1}{n+1}. To simplify this expression, we need to first understand the properties of fractions and how they can be manipulated.

Step 1: Factorize the Numerators and Denominators

The first step in simplifying the expression is to factorize the numerators and denominators of each fraction. The numerator of the first fraction, 4n4n, can be factored as 4n=4⋅n4n = 4 \cdot n. The denominator of the first fraction, 4n−44n-4, can be factored as 4n−4=4(n−1)4n-4 = 4(n-1). The numerator of the second fraction, n−1n-1, is already factored. The denominator of the second fraction, n+1n+1, is also already factored.

$\frac{4n}{4n-4} = \frac{4 \cdot n}{4(n-1)}$
$\frac{n-1}{n+1} = \frac{n-1}{n+1}$

Step 2: Cancel Out Common Factors

Now that we have factored the numerators and denominators, we can cancel out common factors between the two fractions. The common factor between the two fractions is n−1n-1. We can cancel out this factor by dividing both the numerator and denominator of the first fraction by n−1n-1.

$\frac{4 \cdot n}{4(n-1)} \cdot \frac{n-1}{n+1} = \frac{4 \cdot n}{4} \cdot \frac{1}{n+1}$

Step 3: Simplify the Expression

Now that we have canceled out the common factor, we can simplify the expression further. We can simplify the numerator of the first fraction by dividing 4⋅n4 \cdot n by 44, which gives us nn.

$\frac{4 \cdot n}{4} \cdot \frac{1}{n+1} = n \cdot \frac{1}{n+1}$

Step 4: Write the Final Answer

The final simplified expression is n⋅1n+1n \cdot \frac{1}{n+1}. We can write this expression as nn+1\frac{n}{n+1}.

Conclusion

In this article, we have simplified the algebraic expression 4n4n−4⋅n−1n+1\frac{4n}{4n-4} \cdot \frac{n-1}{n+1} by following a step-by-step approach. We have factorized the numerators and denominators, canceled out common factors, and simplified the expression to obtain the final answer, which is nn+1\frac{n}{n+1}. This expression is one of the options provided in the discussion category, and we have shown that it is the correct answer.

Answer

The correct answer is B. nn+1\frac{n}{n+1}.

Additional Tips and Resources

  • To simplify algebraic expressions, it is essential to understand the properties of fractions and how they can be manipulated.
  • Factorizing the numerators and denominators is a crucial step in simplifying algebraic expressions.
  • Canceling out common factors can help simplify the expression further.
  • Practice simplifying different types of algebraic expressions to become proficient in this skill.

References

Q: What is an algebraic expression?

A: An algebraic expression is a mathematical expression that consists of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division.

Q: Why is it important to simplify algebraic expressions?

A: Simplifying algebraic expressions is essential in mathematics because it helps to:

  • Reduce the complexity of the expression
  • Make it easier to solve equations and inequalities
  • Identify patterns and relationships between variables
  • Make calculations more efficient

Q: What are the steps involved in simplifying an algebraic expression?

A: The steps involved in simplifying an algebraic expression are:

  1. Factorize the numerators and denominators
  2. Cancel out common factors
  3. Simplify the expression

Q: How do I factorize the numerators and denominators?

A: To factorize the numerators and denominators, you need to identify the common factors and express them as a product of prime factors.

Q: What is a common factor?

A: A common factor is a factor that appears in both the numerator and denominator of a fraction.

Q: How do I cancel out common factors?

A: To cancel out common factors, you need to divide both the numerator and denominator by the common factor.

Q: What is the final simplified expression?

A: The final simplified expression is the result of canceling out common factors and simplifying the expression.

Q: Can I simplify an algebraic expression with multiple variables?

A: Yes, you can simplify an algebraic expression with multiple variables by following the same steps as before.

Q: What are some common algebraic expressions that can be simplified?

A: Some common algebraic expressions that can be simplified include:

  • 4n4n−4⋅n−1n+1\frac{4n}{4n-4} \cdot \frac{n-1}{n+1}
  • 2x+3x−2\frac{2x+3}{x-2}
  • x2+4x+4x+2\frac{x^2+4x+4}{x+2}

Q: How do I know if an algebraic expression can be simplified?

A: You can determine if an algebraic expression can be simplified by looking for common factors between the numerator and denominator.

Q: What are some real-world applications of simplifying algebraic expressions?

A: Simplifying algebraic expressions has many real-world applications, including:

  • Physics: Simplifying algebraic expressions is essential in physics to describe the motion of objects and the behavior of physical systems.
  • Engineering: Simplifying algebraic expressions is crucial in engineering to design and optimize systems.
  • Economics: Simplifying algebraic expressions is important in economics to model and analyze economic systems.

Conclusion

Simplifying algebraic expressions is a fundamental skill in mathematics that has many real-world applications. By following the steps involved in simplifying an algebraic expression, you can reduce the complexity of the expression, make it easier to solve equations and inequalities, and identify patterns and relationships between variables.