Simplifying Algebraic Expressions A Step-by-Step Guide

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to represent complex equations in a more manageable and understandable form. This is especially crucial in algebra, where expressions often involve variables and exponents. When you want to simplify algebraic expressions completely, there are some important steps that you should follow. This article will guide you through the process of simplifying the expression (-7y^13 - 21y^11 - 28y^7) / (7y^3) completely, offering a detailed, step-by-step explanation to enhance your understanding. Mastering this skill is not only essential for academic success but also for various real-world applications where mathematical models are used. By learning how to break down and simplify expressions, you gain a powerful tool for problem-solving and critical thinking. So, let’s dive into the specifics of this particular problem and learn how to tackle it efficiently.

The initial expression we need to simplify is a fraction where the numerator is a polynomial and the denominator is a monomial. The numerator consists of three terms, each involving the variable 'y' raised to different powers, and the denominator is a single term with 'y' raised to the power of 3. The key to simplifying this expression lies in recognizing the common factors and applying the rules of exponents. We will break down the expression into smaller, more manageable parts, making it easier to identify common factors and simplify each term individually. This approach not only simplifies the process but also helps in understanding the underlying principles of algebraic simplification. Throughout this guide, we will highlight the rules and techniques used, ensuring that you not only get the answer but also grasp the concepts involved. By the end of this article, you will be equipped with the knowledge and skills to simplify similar expressions with confidence. Remember, practice is key to mastering these concepts, so feel free to try more examples and reinforce your understanding.

The first step in simplifying the expression $\frac{-7 y^{13}-21 y^{11}-28 y^7}{7 y^3}$ is to identify and factor out the greatest common factor (GCF) from the numerator. In this case, the numerator is -7y^13 - 21y^11 - 28y^7. To find the GCF, we need to consider both the numerical coefficients and the variable parts of each term. For the coefficients (-7, -21, and -28), the GCF is -7. This is because -7 is the largest number that divides evenly into all three coefficients. Now, let’s look at the variable parts: y^13, y^11, and y^7. The GCF for these terms is the variable 'y' raised to the lowest power present in the terms, which is y^7. Therefore, the greatest common factor for the entire numerator is -7y^7.

Factoring out -7y^7 from the numerator, we get -7y^7(y6 + 3y^4 + 4). This means we divide each term in the numerator by -7y^7 and write the result inside the parentheses. When we divide -7y^13 by -7y^7, we get y^6. When we divide -21y^11 by -7y^7, we get 3y^4. And when we divide -28y^7 by -7y^7, we get 4. So, the factored form of the numerator is indeed -7y^7(y6 + 3y^4 + 4). This step is crucial because it simplifies the expression by isolating the common factors. By factoring out the GCF, we make it easier to see what can be canceled out with the denominator later on. This technique is a cornerstone of simplifying algebraic expressions and is widely used in various mathematical contexts. Recognizing and extracting the GCF not only simplifies the expression but also provides a clearer view of the underlying structure of the polynomial. Mastering this skill will significantly enhance your ability to manipulate and simplify algebraic expressions.

After factoring out the greatest common factor (GCF) from the numerator in the previous step, we can rewrite the original expression. The original expression was $\frac{-7 y^{13}-21 y^{11}-28 y^7}{7 y^3}$. Now that we have factored the numerator as -7y^7(y6 + 3y^4 + 4), we can rewrite the expression as $\frac{-7y^7(y^6 + 3y^4 + 4)}{7y^3}$. This rewritten form is crucial because it sets the stage for simplifying the expression further by canceling out common factors between the numerator and the denominator. The act of rewriting the expression with the factored numerator is more than just a cosmetic change; it highlights the common factors and makes them readily available for simplification. This step is a bridge between identifying the GCF and actually simplifying the expression, making the process more transparent and easier to follow. By clearly presenting the factored form in the numerator, we prepare the expression for the next step, which involves canceling out the common factors with the denominator. This approach is a fundamental technique in algebraic simplification, and mastering it will allow you to handle complex expressions with greater ease and confidence.

The rewritten expression $\frac{-7y^7(y^6 + 3y^4 + 4)}{7y^3}$ now clearly shows the factored form in the numerator, which includes the GCF -7y^7 and the remaining polynomial (y^6 + 3y^4 + 4). The denominator remains as 7y^3. This setup is ideal for the next step, where we will identify and cancel out the common factors between the numerator and the denominator. The rewritten expression not only simplifies the process visually but also reinforces the importance of factoring in algebraic simplification. By breaking down the expression into its constituent factors, we gain a better understanding of its structure and how it can be further simplified. This technique is widely applicable in various mathematical contexts, from solving equations to simplifying complex fractions. The ability to rewrite expressions in this manner is a valuable skill that will enhance your problem-solving capabilities in algebra and beyond. By taking this intermediate step, we ensure a clearer and more methodical approach to simplifying the given expression.

In this step, we focus on canceling out the common factors between the numerator and the denominator of the expression $\frac{-7y^7(y^6 + 3y^4 + 4)}{7y^3}$. We have two components to consider: the numerical coefficients and the variable parts. First, let’s look at the coefficients. In the numerator, we have -7, and in the denominator, we have 7. The common factor between -7 and 7 is 7. When we divide -7 by 7, we get -1, and when we divide 7 by 7, we get 1. So, the numerical part of the expression simplifies to -1.

Next, we consider the variable parts. In the numerator, we have y^7, and in the denominator, we have y^3. To simplify these, we use the rule of exponents which states that when dividing terms with the same base, we subtract the exponents. Therefore, y^7 divided by y^3 is y^(7-3) = y^4. Now, we combine the simplified numerical coefficient and variable part. We have -1 from the numerical part and y^4 from the variable part. Thus, -7y^7 divided by 7y^3 simplifies to -y^4. This process of canceling common factors is a crucial technique in simplifying algebraic expressions. By identifying and removing these factors, we reduce the expression to its simplest form, making it easier to work with and understand. The remaining polynomial (y^6 + 3y^4 + 4) in the numerator does not have any common factors with the simplified denominator (which is now 1), so it remains unchanged. This step is not only about finding the correct answer but also about understanding the underlying mathematical principles that govern algebraic simplification. By mastering this technique, you will be able to tackle a wide range of algebraic problems with greater confidence and efficiency.

After canceling out the common factors in the previous step, we are now ready to write the simplified expression. From Step 3, we found that $\frac{-7y^7}{7y^3}$ simplifies to -y^4. The remaining part of the numerator, (y^6 + 3y^4 + 4), was not affected by the cancellation since it had no common factors with the denominator once the GCF was removed. Therefore, the simplified expression is obtained by multiplying -y^4 with the remaining polynomial. This means we distribute -y^4 across each term inside the parentheses (y^6 + 3y^4 + 4). When we multiply -y^4 by y^6, we add the exponents, resulting in -y^(4+6) = -y^10. Next, we multiply -y^4 by 3y^4. Again, we add the exponents, resulting in -3y^(4+4) = -3y^8. Finally, we multiply -y^4 by 4, which gives us -4y^4.

Combining these terms, the simplified expression is -y^10 - 3y^8 - 4y^4. This is the final simplified form of the original expression. Writing the simplified expression involves not only performing the necessary arithmetic operations but also presenting the result in a clear and concise manner. The simplified expression is now in a form that is easier to understand and work with, which is the ultimate goal of simplifying algebraic expressions. This step is the culmination of all the previous steps, where we identified the GCF, factored the numerator, canceled out common factors, and performed the final distribution. The resulting simplified expression is a more manageable representation of the original problem, making it easier to analyze and use in further calculations or applications. This final step reinforces the importance of attention to detail and accuracy in algebraic manipulations, ensuring that the simplified form is both correct and clearly presented.

In conclusion, by following a step-by-step approach, we have successfully simplified the expression $\frac{-7 y^{13}-21 y^{11}-28 y^7}{7 y^3}$. We began by identifying and factoring out the greatest common factor (GCF) from the numerator, which was -7y^7. This allowed us to rewrite the expression in a more manageable form, highlighting the common factors between the numerator and the denominator. Next, we canceled out these common factors, simplifying both the numerical coefficients and the variable parts. Finally, we distributed the remaining terms to arrive at the final simplified expression. The simplified expression is -y^10 - 3y^8 - 4y^4. This result represents the original expression in its most basic form, making it easier to understand and use in further mathematical operations or applications. Throughout this process, we emphasized the importance of understanding the underlying principles of algebraic simplification, such as factoring, canceling common factors, and applying the rules of exponents.

The ability to simplify algebraic expressions is a fundamental skill in mathematics, with applications in various fields, including science, engineering, and economics. Mastering these techniques not only enhances your problem-solving abilities but also provides a solid foundation for more advanced mathematical concepts. The step-by-step approach we used in this guide can be applied to a wide range of algebraic simplification problems. Remember, the key to success in mathematics is practice. By working through various examples and applying the techniques learned, you can build confidence and proficiency in simplifying algebraic expressions. This final simplified expression is not just an answer; it represents a journey through algebraic manipulation, highlighting the importance of each step in the process. The skills and knowledge gained from this exercise will serve you well in your mathematical endeavors, enabling you to tackle complex problems with greater ease and accuracy. The simplified form -y^10 - 3y^8 - 4y^4 demonstrates the power of algebraic simplification in making complex expressions more accessible and manageable.