Simplifying Algebraic Fractions A Step-by-Step Solution For (-20t⁵u²v³)/(48t⁷u⁴v)

In the realm of mathematics, simplifying fractions is a fundamental skill, especially when dealing with algebraic expressions involving exponents. This article will provide a comprehensive, step-by-step guide on how to reduce the fraction (-20t⁵u²v³)/(48t⁷u⁴v). We will break down the process into manageable parts, ensuring a clear understanding of each step. By mastering this technique, you'll be able to tackle more complex algebraic simplifications with confidence. Understanding the core principles of simplifying fractions and exponents is crucial for success in algebra and beyond. This article aims to not only provide the solution but also to illuminate the underlying concepts, making the process intuitive and reproducible.

Understanding the Basics of Fraction Simplification

Before diving into the specific problem, it's essential to grasp the basics of fraction simplification. At its heart, simplifying a fraction means reducing it to its lowest terms. This involves dividing both the numerator (the top part of the fraction) and the denominator (the bottom part) by their greatest common divisor (GCD). The greatest common divisor is the largest number that divides both the numerator and the denominator without leaving a remainder. When dealing with algebraic expressions, we extend this concept to include variables and their exponents. Simplifying algebraic fractions requires a firm understanding of exponent rules, particularly the quotient rule, which states that when dividing like bases, you subtract the exponents. For example, xᵃ / xᵇ = xᵃ⁻ᵇ. This rule is pivotal in simplifying expressions where variables are raised to different powers. Furthermore, simplifying fractions involves identifying common factors in both the numerator and denominator and canceling them out. This process reduces the fraction to its simplest form, making it easier to work with in subsequent calculations. The ability to simplify fractions efficiently is not just a mathematical skill; it's a foundational element for problem-solving in various scientific and engineering disciplines. Simplifying fractions makes complex equations more manageable, reveals underlying relationships, and ultimately enhances mathematical fluency.

Step 1: Simplify the Numerical Coefficients

The first step in simplifying the fraction (-20t⁵u²v³)/(48t⁷u⁴v) is to focus on the numerical coefficients, which are -20 and 48. To simplify these, we need to find their greatest common divisor (GCD). The GCD of 20 and 48 is 4. This means that both 20 and 48 can be divided evenly by 4. Dividing -20 by 4 gives us -5, and dividing 48 by 4 gives us 12. So, the fraction of the coefficients simplifies from -20/48 to -5/12. This simplification reduces the complexity of the fraction, making it easier to manage the variable terms in the subsequent steps. Finding the GCD is a crucial part of simplifying numerical fractions, as it ensures that the fraction is reduced to its lowest terms. In this case, identifying 4 as the GCD allows us to simplify the coefficients effectively. This step is not only about finding the numbers that divide evenly; it's about understanding the fundamental relationship between the numerator and denominator. By reducing the coefficients first, we set the stage for simplifying the variable components of the fraction, ultimately leading to a cleaner and more concise expression.

Step 2: Simplify the Variable 't'

Now, let's turn our attention to the variable 't'. In the fraction (-20t⁵u²v³)/(48t⁷u⁴v), 't' appears with exponents of 5 in the numerator (t⁵) and 7 in the denominator (t⁷). To simplify this, we apply the quotient rule of exponents, which states that when dividing like bases, you subtract the exponents. So, t⁵ / t⁷ becomes t⁵⁻⁷, which simplifies to t⁻². A negative exponent indicates that the variable should be moved to the denominator to make the exponent positive. Therefore, t⁻² is equivalent to 1/t². This means that the 't' terms in the fraction simplify to 1/t², indicating that t² remains in the denominator. This step is crucial for simplifying algebraic fractions, as it eliminates common factors of 't' from the numerator and denominator. The use of the quotient rule is a fundamental technique in algebra, allowing us to reduce complex expressions involving exponents. By correctly applying this rule, we ensure that the variable 't' is simplified to its lowest terms, contributing to the overall simplification of the fraction. Understanding how to manipulate exponents is essential for success in algebra and calculus, making this step a key part of the simplification process.

Step 3: Simplify the Variable 'u'

Next, we focus on the variable 'u' in the fraction (-20t⁵u²v³)/(48t⁷u⁴v). 'u' has an exponent of 2 in the numerator (u²) and an exponent of 4 in the denominator (u⁴). Applying the quotient rule of exponents again, we subtract the exponents: u² / u⁴ becomes u²⁻⁴, which simplifies to u⁻². Similar to the 't' variable, the negative exponent indicates that the variable should be moved to the denominator to make the exponent positive. Thus, u⁻² is equivalent to 1/u². This means that the 'u' terms simplify to 1/u², indicating that u² remains in the denominator. This step further reduces the complexity of the fraction by eliminating common factors of 'u' from the numerator and denominator. The consistent application of the quotient rule ensures that we handle the exponents correctly, leading to an accurate simplification. Simplifying the 'u' terms is essential for achieving the fraction's lowest form, making it easier to work with in future calculations. Understanding and applying these exponent rules is a cornerstone of algebraic manipulation, crucial for simplifying various mathematical expressions.

Step 4: Simplify the Variable 'v'

Now, let's simplify the variable 'v' in the fraction (-20t⁵u²v³)/(48t⁷u⁴v). 'v' has an exponent of 3 in the numerator (v³) and an exponent of 1 in the denominator (v, which is the same as v¹). Applying the quotient rule of exponents, we subtract the exponents: v³ / v¹ becomes v³⁻¹, which simplifies to v². In this case, the resulting exponent is positive, so v² remains in the numerator. This step finalizes the simplification of the variable terms in the fraction. By correctly applying the quotient rule, we ensure that 'v' is simplified to its lowest terms, contributing significantly to the overall reduction of the fraction. This step highlights the importance of understanding exponent rules in simplifying algebraic expressions. The ability to manipulate exponents and variables is crucial for solving more complex mathematical problems. By simplifying the 'v' terms, we bring the fraction closer to its simplest form, making it easier to interpret and use in further calculations.

Step 5: Combine the Simplified Terms

Finally, after simplifying the numerical coefficients and each variable separately, we combine the simplified terms to obtain the reduced fraction. From Step 1, we simplified the coefficients to -5/12. From Step 2, the 't' terms simplified to 1/t². From Step 3, the 'u' terms simplified to 1/u². And from Step 4, the 'v' terms simplified to v². Now, we combine these simplified terms: (-5/12) * (1/t²) * (1/u²) * v². Multiplying these together, we get (-5 * 1 * 1 * v²) / (12 * t² * u²), which simplifies to -5v² / (12t²u²). This is the simplified form of the original fraction. By combining the simplified terms, we arrive at a concise and manageable expression. This final step demonstrates the importance of breaking down a complex problem into smaller, more manageable parts. Each simplification step contributes to the overall reduction of the fraction, leading to the final answer. The ability to combine these simplified terms effectively showcases a strong understanding of algebraic manipulation and fraction simplification.

Final Answer: -5v² / (12t²u²)

Therefore, the simplified form of the fraction (-20t⁵u²v³)/(48t⁷u⁴v) is -5v² / (12t²u²). This result matches option B) from the original question. Through a step-by-step process, we have demonstrated how to simplify a complex algebraic fraction by addressing each component individually: the numerical coefficients, and each variable with its respective exponent. By finding the greatest common divisor of the coefficients and applying the quotient rule of exponents, we systematically reduced the fraction to its simplest form. This process underscores the importance of understanding fundamental algebraic principles and applying them methodically to solve problems. Simplifying fractions is not just a mathematical exercise; it is a skill that enhances problem-solving abilities and provides a foundation for more advanced mathematical concepts. The final answer, -5v² / (12t²u²), is a testament to the power of systematic simplification and the clarity it brings to mathematical expressions. Mastering these techniques is crucial for success in algebra and beyond, enabling confident navigation of complex equations and problems.