Simplifying Expressions With Exponents A Step By Step Guide

This article delves into simplifying algebraic expressions, specifically focusing on how to handle negative exponents and the rules of division involving variables. We'll break down the process step-by-step, ensuring a clear understanding of the underlying principles. Our central problem involves simplifying the expression $\frac{10 x^6 y^{12}}{-5 x^{-2} y^{-6}}$, where $x$ and $y$ are non-zero. This type of problem is a cornerstone of algebra, appearing in various contexts, from basic equation solving to more advanced calculus. Mastering the simplification of expressions with exponents is crucial for success in mathematics. We will explore the rules governing exponents, particularly negative exponents and the quotient rule, to arrive at the simplified form. Understanding these rules will not only help solve this specific problem but also equip you with the tools to tackle a wide range of algebraic simplifications. We'll then compare our simplified result with the provided options to determine the correct answer.

Understanding the Fundamentals of Exponents

Before we dive into the solution, let's recap the fundamental rules of exponents. These rules are the building blocks for simplifying any expression involving powers. The most basic rule is that $x^n$ means multiplying $x$ by itself $n$ times. For example, $x^3 = x \cdot x \cdot x$. A crucial concept for our problem is the negative exponent rule: $x^{-n} = \frac{1}{x^n}$. This rule states that a variable raised to a negative power is equivalent to the reciprocal of the variable raised to the positive power. For instance, $x^{-2}$ is the same as $\frac{1}{x^2}$. This is the key to handling the negative exponents in our expression. Another important rule is the quotient rule, which states that when dividing exponential expressions with the same base, you subtract the exponents: $\frac{x^m}{x^n} = x^{m-n}$. This rule applies directly to the $x$ and $y$ terms in our expression. We also need to remember the rules of dividing coefficients. When dividing terms with numerical coefficients, we simply perform the division operation as usual. These rules, when applied correctly, will lead us to the simplified form of the given expression. A firm grasp of these exponent rules is essential for anyone studying algebra and beyond. By understanding how exponents work, we can manipulate and simplify complex expressions, making them easier to work with and understand.

Step-by-Step Simplification of the Expression

Now, let's apply these rules to simplify the expression $\frac{10 x^6 y^{12}}{-5 x^{-2} y^{-6}}$. First, we can separate the numerical coefficients and the variable terms: $\frac{10}{-5} \cdot \frac{x^6}{x^{-2}} \cdot \frac{y^{12}}{y^{-6}}$. Dividing the coefficients, we get $\frac{10}{-5} = -2$. Next, we apply the quotient rule to the $x$ terms: $\frac{x^6}{x^{-2}} = x^{6 - (-2)} = x^{6 + 2} = x^8$. Remember, subtracting a negative number is the same as adding its positive counterpart. Similarly, for the $y$ terms, we have $\frac{y^{12}}{y^{-6}} = y^{12 - (-6)} = y^{12 + 6} = y^{18}$. Putting it all together, we have $-2 \cdot x^8 \cdot y^{18}$, which simplifies to $-2x^8y^{18}$. This step-by-step breakdown shows how each rule of exponents is applied to arrive at the final simplified expression. By isolating each component (coefficients and variables), we can apply the appropriate rule without making errors. This method ensures a clear and organized approach to simplifying complex expressions. It's important to practice these steps to become comfortable with manipulating exponents and to avoid common mistakes. The result we have obtained, $-2x^8y^{18}$, now needs to be compared with the given options to identify the correct answer.

Matching the Simplified Expression with the Options

Having simplified the expression to $-2x^8y^{18}$, we now need to compare this result with the given options: A. $-50 x^8 y^{18}$ B. $-2 x^8 y^{18}$ C. $-2 x^{12} y^{72}$ D. $5 x^8 y^{18}$. By direct comparison, we can see that option B, $-2 x^8 y^{18}$, matches our simplified expression perfectly. The other options differ in either the coefficient or the exponents of the variables. Option A has the wrong coefficient (-50 instead of -2), option C has incorrect exponents for both $x$ and $y$, and option D has the wrong sign for the coefficient (positive 5 instead of negative 2). This comparison highlights the importance of accurate simplification and careful attention to detail. Even a small error in applying the exponent rules or dividing coefficients can lead to an incorrect answer. Therefore, it's always a good idea to double-check your work, especially in problems involving multiple steps. This final step of comparing the simplified expression with the options ensures that we have arrived at the correct answer and reinforces the importance of precision in algebraic manipulations. In this case, option B is the only expression that is equivalent to the original expression.

Conclusion: Mastering Exponent Rules

In conclusion, the expression equivalent to $\frac{10 x^6 y^{12}}{-5 x^{-2} y^{-6}}$ is $-2 x^8 y^{18}$, which corresponds to option B. This problem demonstrates the importance of understanding and applying the rules of exponents, particularly the negative exponent rule and the quotient rule. By breaking down the expression into smaller parts and applying each rule step-by-step, we were able to simplify the expression accurately. This systematic approach is crucial for tackling more complex algebraic problems. Furthermore, carefully comparing the simplified expression with the given options is essential to ensure the correct answer is selected. Mastering these concepts and techniques is fundamental for success in algebra and beyond. Exponents are a core component of many mathematical concepts, and a strong foundation in exponent rules will benefit students in various areas of study. Remember to practice these rules regularly to build fluency and confidence in simplifying algebraic expressions. The ability to manipulate exponents effectively is a valuable skill that will serve you well in your mathematical journey.

Practice Problems

To further solidify your understanding, try simplifying these expressions:

1. $\frac{15 a^4 b^{-3}}{3 a^{-2} b^5}$
2. $\frac{-8 x^{-5} y^7}{2 x^2 y^{-4}}$
3. \frac{24 c^9 d^{-1}}{-6 c^{-3} d^{-8}

By working through these practice problems, you can reinforce your understanding of the exponent rules and develop your problem-solving skills. Remember to break down each expression step-by-step and apply the appropriate rules carefully. Check your answers and review any mistakes to identify areas where you need more practice. Consistent practice is the key to mastering exponents and other algebraic concepts.