Simplifying Algebraic Expressions Divide (12u^7 - 8u^5) By (4u^3)

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Introduction

In the realm of mathematics, simplification is a cornerstone of problem-solving. Complex expressions, when simplified, reveal their underlying structure and become easier to manipulate. This article delves into the process of simplifying the algebraic expression (12u^7 - 8u^5) / (4u^3). Our main keyword, simplification of algebraic expressions, will be central to our exploration as we break down the expression, identify common factors, and apply the rules of exponents to arrive at the most concise form. We will also explore the division of polynomials which is a core concept to master the simplification process. The journey through this simplification will not only provide the solution but also illuminate the fundamental principles that govern algebraic manipulations. Understanding these principles is crucial for tackling more complex mathematical challenges, whether in algebra, calculus, or other branches of mathematics. The ability to simplify expressions is a powerful tool that empowers mathematicians and students alike to approach problems with clarity and confidence.

Breaking Down the Expression

The given expression, (12u^7 - 8u^5) / (4u^3), is a fraction where the numerator is a binomial (a polynomial with two terms) and the denominator is a monomial (a polynomial with one term). To simplify this expression, we will employ the principle of dividing each term in the numerator by the denominator. This approach allows us to address each part of the expression individually, making the simplification process more manageable. The core concept here is the distributive property of division over subtraction. This property states that (a - b) / c = a / c - b / c. Applying this property to our expression, we get: (12u^7 / 4u^3) - (8u^5 / 4u^3). This transformation is a crucial first step in our simplification journey. It breaks down the complex fraction into two simpler fractions that we can tackle independently. The significance of this step lies in its ability to transform a seemingly daunting expression into a series of more manageable sub-problems. This is a common strategy in mathematics: break down complex problems into smaller, more digestible parts. By focusing on each term separately, we can apply the rules of exponents and arithmetic with greater precision and clarity.

Applying the Rules of Exponents

Now that we've separated the expression into two fractions, 12u^7 / 4u^3 and 8u^5 / 4u^3, we can apply the rules of exponents to simplify each term. The key rule here is the quotient rule of exponents, which states that when dividing exponents with the same base, you subtract the powers: x^m / x^n = x^(m-n). Let's apply this rule to the first term, 12u^7 / 4u^3. First, we divide the coefficients: 12 / 4 = 3. Then, we apply the quotient rule to the variable part: u^7 / u^3 = u^(7-3) = u^4. So, the first term simplifies to 3u^4. Now, let's apply the same process to the second term, 8u^5 / 4u^3. Dividing the coefficients, we get 8 / 4 = 2. Applying the quotient rule to the variable part, we have u^5 / u^3 = u^(5-3) = u^2. Thus, the second term simplifies to 2u^2. The application of the quotient rule of exponents is a fundamental technique in simplifying algebraic expressions. It allows us to condense terms with exponents, making the expression more compact and easier to work with. The ability to manipulate exponents is a crucial skill in algebra and beyond.

Combining the Simplified Terms

After applying the rules of exponents, we've simplified the original expression into two terms: 3u^4 and 2u^2. Now, we combine these simplified terms to get the final simplified expression. Recall that our expression was initially broken down into (12u^7 / 4u^3) - (8u^5 / 4u^3), which we then simplified to 3u^4 - 2u^2. Therefore, the final simplified expression is simply 3u^4 - 2u^2. This is the most concise form of the original expression. At this point, it's important to check if further simplification is possible. In this case, we can see that both terms share a common factor of u^2. However, since the question asks us to simplify the expression as much as possible, we will leave it in this polynomial form. The process of combining simplified terms is a crucial step in any simplification problem. It brings together the results of individual simplifications to form the final answer. This step highlights the importance of keeping track of the operations and signs throughout the simplification process.

Factoring out the Greatest Common Factor (Optional, but insightful)

While we have arrived at a simplified form, it's worth exploring an additional step that can sometimes be useful: factoring out the greatest common factor (GCF). In our simplified expression, 3u^4 - 2u^2, we can observe that both terms share a common factor of u^2. We can also factor out the GCF from the coefficients, which in this case is 1 (since 3 and 2 have no common factors other than 1). Therefore, the greatest common factor for the entire expression is u^2. To factor out u^2, we divide each term by u^2 and place the result inside parentheses, with u^2 outside the parentheses: u2(3u2 - 2). This factored form is equivalent to our simplified expression, 3u^4 - 2u^2. While factoring is not strictly required in this case, it can be beneficial in other contexts, such as solving equations or further simplifying more complex expressions. Factoring is a powerful technique that allows us to rewrite expressions in different forms, often revealing hidden relationships and structures. It's an essential tool in the arsenal of any mathematician.

Conclusion

In this article, we successfully simplified the algebraic expression (12u^7 - 8u^5) / (4u^3) to its most concise form: 3u^4 - 2u^2. Our journey involved breaking down the expression, applying the distributive property of division, utilizing the quotient rule of exponents, and combining the simplified terms. We also explored the optional but insightful step of factoring out the greatest common factor. The process of simplifying algebraic expressions is a fundamental skill in mathematics, and the techniques we've discussed here are applicable to a wide range of problems. By mastering these techniques, you'll be well-equipped to tackle more complex mathematical challenges. The ability to simplify expressions is not just about finding the right answer; it's about developing a deeper understanding of mathematical structures and relationships. It's about gaining the confidence to manipulate equations and expressions with precision and clarity. As you continue your mathematical journey, remember that simplification is a powerful tool that can unlock the beauty and elegance of mathematics.

Final Answer

The simplified form of the expression (12u^7 - 8u^5) / (4u^3) is 3u^4 - 2u^2.