Simplifying Algebraic Expressions A Step By Step Guide

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In the realm of mathematics, encountering complex algebraic expressions is a common challenge. This article delves into one such expression, aiming to simplify it and identify its equivalent form. Specifically, we will dissect the expression $\frac{2 d-6}{d^2+2 d-48} \div \frac{d-3}{2 d+16}$, navigating through the steps of factorization, division, and simplification to arrive at the correct answer. Our goal is not just to provide the solution but to offer a comprehensive understanding of the underlying principles, ensuring that readers can confidently tackle similar problems in the future.

Deconstructing the Expression: A Step-by-Step Approach

When faced with a complex expression like the one at hand, the most effective strategy is to break it down into smaller, more manageable parts. This involves identifying opportunities for factorization, simplifying individual fractions, and then performing the division operation. By meticulously working through each step, we can transform the original expression into its simplest equivalent form. Let's embark on this journey, starting with the numerator of the first fraction.

Factoring the Numerator and Denominator

Our first task is to factorize the numerator and denominator of the given expression. This involves identifying common factors and rewriting the expression in a more simplified form. Let's begin with the numerator, which is $2d - 6$. We can observe that both terms have a common factor of 2. Factoring out the 2, we get:

$2d - 6 = 2(d - 3)$

Now, let's move on to the denominator, which is a quadratic expression: $d^2 + 2d - 48$. To factorize this, we need to find two numbers that multiply to -48 and add up to 2. These numbers are 8 and -6. Therefore, we can factorize the denominator as follows:

$d^2 + 2d - 48 = (d + 8)(d - 6)$

Next, consider the second fraction's denominator, $2d + 16$. Here, the common factor is 2, so we can rewrite it as:

$2d + 16 = 2(d + 8)$

Now, rewriting the original expression with these factorizations, we have:

$\frac{2(d - 3)}{(d + 8)(d - 6)} \div \frac{d - 3}{2(d + 8)}$

This factorization step is crucial as it lays the groundwork for simplifying the expression further by canceling out common factors. It's akin to disassembling a complex machine into its constituent parts, making it easier to understand and work with. The ability to factorize algebraic expressions is a fundamental skill in mathematics, and mastering it can significantly enhance problem-solving capabilities.

Dividing Fractions: The Flip and Multiply Technique

In the realm of arithmetic, dividing by a fraction is synonymous with multiplying by its reciprocal. This principle extends seamlessly into the realm of algebraic expressions. When confronted with the task of dividing one fraction by another, we employ the time-honored technique of "flipping" the second fraction and then multiplying. This transformation allows us to convert a division problem into a multiplication problem, which is often more straightforward to handle.

Consider the expression $\frac{A}{B} \div \frac{C}{D}$. To perform this division, we invert the second fraction, $\frac{C}{D}$, to obtain its reciprocal, $\frac{D}{C}$. Then, we multiply the first fraction, $\frac{A}{B}$, by this reciprocal:

$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C} = \frac{A \times D}{B \times C}$

Applying this principle to our expression, $\frac{2(d - 3)}{(d + 8)(d - 6)} \div \frac{d - 3}{2(d + 8)}$, we flip the second fraction and multiply:

$\frac{2(d - 3)}{(d + 8)(d - 6)} \times \frac{2(d + 8)}{d - 3}$

Now, we have a multiplication problem that is poised for simplification. The act of flipping and multiplying is not merely a mechanical procedure; it is a fundamental operation rooted in the principles of fraction manipulation. By understanding this concept, we can confidently navigate through complex expressions and arrive at accurate solutions.

Simplifying the Expression: Canceling Common Factors

Having transformed the division problem into a multiplication problem, the next step is to simplify the expression. This involves identifying and canceling out common factors that appear in both the numerator and the denominator. This process is akin to pruning a tree, removing unnecessary branches to reveal the underlying structure. By eliminating these common factors, we reduce the expression to its most concise and elegant form.

In our expression, $\frac{2(d - 3)}{(d + 8)(d - 6)} \times \frac{2(d + 8)}{d - 3}$, we can observe several common factors. The term $(d - 3)$ appears in both the numerator and the denominator, as does the term $(d + 8)$. These common factors can be canceled out, leaving us with a simplified expression.

$\frac{2 \cancel{(d - 3)}}{\cancel{(d + 8)}(d - 6)} \times \frac{2 \cancel{(d + 8)}}{\cancel{d - 3}} = \frac{2}{d - 6} \times 2$

Multiplying the remaining terms, we get:

$\frac{2 \times 2}{d - 6} = \frac{4}{d - 6}$

This simplification process is not just about arriving at the correct answer; it's about gaining a deeper understanding of the expression's structure and behavior. By canceling out common factors, we reveal the underlying relationships between the terms and obtain a more streamlined representation of the original expression.

Identifying the Correct Answer: Option B

After meticulously working through the steps of factorization, division, and simplification, we have arrived at the equivalent expression: $\frac{4}{d - 6}$. Now, let's compare this result with the options provided in the question:

A. $\frac{d - 3}{d - 6}$ B. $\frac{4}{d - 6}$ C. $\frac{4}{d + 8}$ D. $\frac{2(d + 8)}{d - 3}$

By direct comparison, we can clearly see that our simplified expression, $\frac{4}{d - 6}$, matches option B. Therefore, option B is the correct answer.

This process of identifying the correct answer is not just about selecting the matching option; it's about validating our work and ensuring that we have followed the correct steps. By carefully comparing our result with the given options, we can reinforce our understanding and build confidence in our problem-solving abilities.

Conclusion: Mastering Algebraic Simplification

In this comprehensive guide, we have successfully navigated through the complexities of the algebraic expression $\frac{2 d-6}{d^2+2 d-48} \div \frac{d-3}{2 d+16}$. By meticulously applying the principles of factorization, division, and simplification, we have arrived at the equivalent expression $\frac{4}{d - 6}$, which corresponds to option B.

This journey has not only provided us with the solution to a specific problem but has also reinforced the fundamental concepts of algebraic manipulation. The ability to factorize expressions, divide fractions, and cancel out common factors are essential skills in mathematics. By mastering these techniques, we can confidently tackle a wide range of algebraic challenges.

Moreover, this exercise has highlighted the importance of a systematic and methodical approach to problem-solving. Breaking down complex expressions into smaller, more manageable parts, and working through each step with precision and care, is crucial for achieving accurate results. This approach not only helps us solve problems effectively but also fosters a deeper understanding of the underlying mathematical principles.

As we conclude this exploration, let us carry forward the knowledge and skills we have gained. With a solid foundation in algebraic simplification, we are well-equipped to tackle future mathematical challenges with confidence and competence. Remember, mathematics is not just about finding the right answer; it's about the journey of discovery and the understanding we gain along the way.

This article provides a detailed solution and explanation for simplifying the algebraic expression $\frac{2 d-6}{d^2+2 d-48} \div \frac{d-3}{2 d+16}$. It covers the steps involved in factoring, dividing fractions, and canceling common factors to arrive at the correct answer. This guide is designed to help students and anyone interested in mathematics to better understand algebraic simplification.

Understanding the Algebraic Expression $\frac{2 d-6}{d^2+2 d-48} \div \frac{d-3}{2 d+16}$

In mathematics, algebraic expressions often require simplification to make them easier to understand and work with. This involves a series of steps, including factoring, dividing fractions, and canceling common factors. The expression we are focusing on is $\frac{2 d-6}{d^2+2 d-48} \div \frac{d-3}{2 d+16}$. Simplifying this expression requires a clear understanding of algebraic principles and a methodical approach.

Step 1 Factoring the Numerator and Denominator

Factoring is the first crucial step in simplifying algebraic expressions. It involves breaking down the expressions into their constituent factors. Let's start by factoring the numerator of the first fraction, $2d - 6$. The common factor here is 2, so we can rewrite the numerator as:

$2d - 6 = 2(d - 3)$

Next, we factor the denominator of the first fraction, $d^2 + 2d - 48$. This is a quadratic expression, and we need to find two numbers that multiply to -48 and add up to 2. These numbers are 8 and -6. Thus, we can factor the denominator as:

$d^2 + 2d - 48 = (d + 8)(d - 6)$

Moving on to the second fraction, we factor the denominator $2d + 16$. The common factor here is 2, so we get:

$2d + 16 = 2(d + 8)$

Now, substituting the factored forms into the original expression, we have:

$\frac{2(d - 3)}{(d + 8)(d - 6)} \div \frac{d - 3}{2(d + 8)}$

Step 2 Dividing Fractions

Dividing by a fraction is the same as multiplying by its reciprocal. This means we need to flip the second fraction and change the division sign to a multiplication sign. So, we rewrite the expression as:

$\frac{2(d - 3)}{(d + 8)(d - 6)} \times \frac{2(d + 8)}{d - 3}$

This step transforms the division problem into a multiplication problem, which is easier to handle.

Step 3 Simplifying the Expression

Simplifying involves canceling out common factors from the numerator and the denominator. In our expression, we can see that $(d - 3)$ and $(d + 8)$ appear in both the numerator and the denominator. We can cancel these out:

$\frac{2 \cancel{(d - 3)}}{\cancel{(d + 8)}(d - 6)} \times \frac{2 \cancel{(d + 8)}}{\cancel{d - 3}} = \frac{2}{d - 6} \times 2$

Multiplying the remaining terms gives us:

$\frac{2 \times 2}{d - 6} = \frac{4}{d - 6}$

Thus, the simplified form of the original expression is $\frac{4}{d - 6}$.

Identifying the Correct Answer

Now that we have simplified the expression, we can identify the correct answer from the given options:

A. $\frac{d - 3}{d - 6}$ B. $\frac{4}{d - 6}$ C. $\frac{4}{d + 8}$ D. $\frac{2(d + 8)}{d - 3}$

Our simplified expression $\frac{4}{d - 6}$ matches option B, so option B is the correct answer.

Conclusion Mastering Algebraic Simplification Techniques

Simplifying algebraic expressions is a fundamental skill in mathematics. By following a systematic approach, we can break down complex expressions into manageable parts and arrive at the correct solution. In this article, we have demonstrated how to simplify the expression $\frac{2 d-6}{d^2+2 d-48} \div \frac{d-3}{2 d+16}$ by factoring, dividing fractions, and canceling common factors. The correct answer is $\frac{4}{d - 6}$, which corresponds to option B.

This process not only helps in solving specific problems but also enhances overall understanding of algebraic principles. Consistent practice and a clear grasp of these techniques are essential for success in mathematics.

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This article offers a comprehensive guide on simplifying the algebraic expression $\frac{2 d-6}{d^2+2 d-48} \div \frac{d-3}{2 d+16}$. We will walk through each step, explaining the mathematical principles involved, to help you understand the simplification process thoroughly. This guide is perfect for students and anyone looking to improve their algebra skills.

Breaking Down the Expression $\frac{2 d-6}{d^2+2 d-48} \div \frac{d-3}{2 d+16}$

Algebraic expressions can seem daunting at first, but with a step-by-step approach, they can be easily simplified. Our focus is on the expression $\frac{2 d-6}{d^2+2 d-48} \div \frac{d-3}{2 d+16}$. To simplify this, we need to factorize, divide fractions, and cancel out common factors. Let's start with factorization.

Factoring the Components

Factoring is the process of breaking down expressions into their multiplicative components. This helps in identifying common factors that can be canceled out later. We will factor the numerator and denominator of both fractions.

Factoring the First Fraction

1. Numerator: $2d - 6$
   • The common factor is 2, so we factor it out: $2(d - 3)$.
2. Denominator: $d^2 + 2d - 48$
   • This is a quadratic expression. We look for two numbers that multiply to -48 and add to 2. These numbers are 8 and -6.
   • So, the factored form is $(d + 8)(d - 6)$.

Factoring the Second Fraction

1. Numerator: $d - 3$ (already in simplest form).
2. Denominator: $2d + 16$
   • The common factor is 2, so we factor it out: $2(d + 8)$.

After factoring, the expression becomes:

$\frac{2(d - 3)}{(d + 8)(d - 6)} \div \frac{d - 3}{2(d + 8)}$

Dividing Fractions Explained

Dividing fractions involves multiplying by the reciprocal of the divisor. In other words, we flip the second fraction and multiply. This is a fundamental rule in algebra and arithmetic.

Applying this rule to our expression, we get:

$\frac{2(d - 3)}{(d + 8)(d - 6)} \times \frac{2(d + 8)}{d - 3}$

Simplifying by Canceling Common Factors

Simplification involves identifying and canceling out common factors in the numerator and denominator. This reduces the expression to its simplest form.

In our expression, we can cancel out $(d - 3)$ and $(d + 8)$:

$\frac{2 \cancel{(d - 3)}}{\cancel{(d + 8)}(d - 6)} \times \frac{2 \cancel{(d + 8)}}{\cancel{(d - 3)}}$

This leaves us with:

$\frac{2}{d - 6} \times 2$

Multiplying the remaining terms gives us:

$\frac{4}{d - 6}$

Choosing the Right Answer

Now that we have simplified the expression, let's compare our result with the given options:

A. $\frac{d - 3}{d - 6}$ B. $\frac{4}{d - 6}$ C. $\frac{4}{d + 8}$ D. $\frac{2(d + 8)}{d - 3}$

Our simplified expression $\frac{4}{d - 6}$ matches option B. Therefore, option B is the correct answer.

Summing Up Key Techniques for Algebraic Simplification

Simplifying algebraic expressions involves a combination of factoring, dividing fractions, and canceling common factors. The key steps include:

1. Factoring: Breaking down expressions into multiplicative components.
2. Dividing Fractions: Multiplying by the reciprocal of the divisor.
3. Simplifying: Canceling common factors to reduce the expression to its simplest form.

By following these steps methodically, you can confidently simplify complex algebraic expressions. The correct answer to the given expression is $\frac{4}{d - 6}$, which corresponds to option B.

This guide provides a detailed explanation of the simplification process, ensuring that you not only arrive at the correct answer but also understand the underlying principles. Practice these techniques regularly to enhance your algebra skills and tackle more complex problems with ease.