Simplify The Expression:${ \frac{5}{2x - 8} + \frac{3x}{x^2 - 16} }$

by ADMIN 70 views

Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and mathematician should possess. In this article, we will delve into the world of algebra and explore the steps involved in simplifying a given expression. We will use the expression 52xβˆ’8+3xx2βˆ’16\frac{5}{2x - 8} + \frac{3x}{x^2 - 16} as a case study and demonstrate how to simplify it using various algebraic techniques.

Understanding the Expression

Before we begin simplifying the expression, let's first understand what it represents. The given expression is a sum of two fractions, each with a different denominator. The first fraction has a denominator of 2xβˆ’82x - 8, while the second fraction has a denominator of x2βˆ’16x^2 - 16. Our goal is to simplify this expression by combining the two fractions into a single fraction with a common denominator.

Step 1: Factor the Denominators

To simplify the expression, we need to factor the denominators of both fractions. The first denominator, 2xβˆ’82x - 8, can be factored as 2(xβˆ’4)2(x - 4). The second denominator, x2βˆ’16x^2 - 16, can be factored as (x+4)(xβˆ’4)(x + 4)(x - 4).

# Factored Denominators
## First Denominator
$2(x - 4)$

Second Denominator


(x+4)(xβˆ’4)(x + 4)(x - 4)

Step 2: Find the Least Common Multiple (LCM)

Now that we have factored the denominators, we need to find the least common multiple (LCM) of the two expressions. The LCM is the smallest expression that both 2(xβˆ’4)2(x - 4) and (x+4)(xβˆ’4)(x + 4)(x - 4) can divide into evenly. In this case, the LCM is (x+4)(xβˆ’4)(x + 4)(x - 4).

# Least Common Multiple (LCM)
## LCM of $2(x - 4)$ and $(x + 4)(x - 4)$
$(x + 4)(x - 4)$

Step 3: Rewrite the Fractions with the LCM

Now that we have found the LCM, we can rewrite both fractions with the LCM as the denominator. We will multiply the numerator and denominator of the first fraction by (x+4)(x + 4), and the numerator and denominator of the second fraction by 22.

# Rewritten Fractions
## First Fraction
$\frac{5(x + 4)}{2(x - 4)(x + 4)}$

Second Fraction


6x2(xβˆ’4)(x+4)\frac{6x}{2(x - 4)(x + 4)}

Step 4: Combine the Fractions

Now that we have rewritten both fractions with the LCM as the denominator, we can combine them into a single fraction. We will add the numerators and keep the common denominator.

# Combined Fraction
$\frac{5(x + 4) + 6x}{2(x - 4)(x + 4)}$

Step 5: Simplify the Numerator

The final step is to simplify the numerator of the combined fraction. We can combine like terms and simplify the expression.

# Simplified Numerator
$\frac{5x + 20 + 6x}{2(x - 4)(x + 4)}$

11x+202(xβˆ’4)(x+4)\frac{11x + 20}{2(x - 4)(x + 4)}

Conclusion

In this article, we have demonstrated how to simplify the expression 52xβˆ’8+3xx2βˆ’16\frac{5}{2x - 8} + \frac{3x}{x^2 - 16} using various algebraic techniques. We have factored the denominators, found the least common multiple (LCM), rewritten the fractions with the LCM, combined the fractions, and simplified the numerator. The final simplified expression is 11x+202(xβˆ’4)(x+4)\frac{11x + 20}{2(x - 4)(x + 4)}. This expression represents the simplified form of the original expression, and it can be used in various mathematical applications.

Final Answer

The final answer is 11x+202(xβˆ’4)(x+4)\boxed{\frac{11x + 20}{2(x - 4)(x + 4)}}.

Introduction

In our previous article, we explored the steps involved in simplifying the expression 52xβˆ’8+3xx2βˆ’16\frac{5}{2x - 8} + \frac{3x}{x^2 - 16}. We demonstrated how to factor the denominators, find the least common multiple (LCM), rewrite the fractions with the LCM, combine the fractions, and simplify the numerator. In this article, we will address some common questions and concerns that students and mathematicians may have when simplifying expressions.

Q&A

Q: What is the purpose of factoring the denominators?

A: Factoring the denominators is an essential step in simplifying expressions. It allows us to identify the common factors between the two denominators and find the least common multiple (LCM). The LCM is the smallest expression that both denominators can divide into evenly.

Q: How do I find the least common multiple (LCM)?

A: To find the LCM, we need to identify the common factors between the two denominators. We can do this by factoring the denominators and then finding the product of the highest power of each common factor.

Q: What is the difference between the least common multiple (LCM) and the greatest common divisor (GCD)?

A: The LCM is the smallest expression that both denominators can divide into evenly, while the GCD is the largest expression that both denominators can divide into evenly. In other words, the LCM is the product of the highest power of each common factor, while the GCD is the product of the lowest power of each common factor.

Q: How do I rewrite the fractions with the LCM?

A: To rewrite the fractions with the LCM, we need to multiply the numerator and denominator of each fraction by the necessary factors to obtain the LCM as the denominator.

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

A: Yes, you can still simplify the expression even if the denominators are not factorable. In this case, you can use other algebraic techniques, such as multiplying the numerator and denominator by a conjugate or using a different method to simplify the expression.

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

A: Some common mistakes to avoid when simplifying expressions include:

  • Not factoring the denominators
  • Not finding the least common multiple (LCM)
  • Not rewriting the fractions with the LCM
  • Not combining the fractions correctly
  • Not simplifying the numerator

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

A: To determine if the expression is simplified, you can check the following:

  • Are the denominators factored?
  • Is the least common multiple (LCM) found?
  • Are the fractions rewritten with the LCM?
  • Are the fractions combined correctly?
  • Is the numerator simplified?

If you have checked all of these conditions and the expression still appears complex, it may not be simplified.

Conclusion

In this article, we have addressed some common questions and concerns that students and mathematicians may have when simplifying expressions. We have discussed the importance of factoring the denominators, finding the least common multiple (LCM), rewriting the fractions with the LCM, combining the fractions, and simplifying the numerator. By following these steps and avoiding common mistakes, you can simplify expressions and arrive at the final answer.

Final Answer

The final answer is 11x+202(xβˆ’4)(x+4)\boxed{\frac{11x + 20}{2(x - 4)(x + 4)}}.