Simplifying Expressions A Step-by-Step Guide To Simplify (-8x^3y)^2

In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex equations and reduce them to their most basic form, making them easier to understand and manipulate. When dealing with expressions involving exponents, variables, and coefficients, the process of simplification often involves applying the rules of exponents and the distributive property. The given expression, (-8x^3y)2, presents an excellent opportunity to illustrate these principles. To effectively simplify this expression, we will need to understand the power of a product rule, which states that (ab)^n = a^n * b^n, and the power of a power rule, which states that (a^m)n = a^(m*n). These rules are the cornerstones of simplifying expressions with exponents, allowing us to distribute the outer exponent across the terms inside the parentheses. The careful application of these rules, combined with a solid understanding of arithmetic operations, will lead us to the simplified form of the expression. By mastering the techniques involved in simplifying expressions like this, students can build a strong foundation for more advanced topics in algebra and calculus. This skill is not only crucial for academic success but also for problem-solving in various real-world scenarios, where complex situations can be modeled and simplified using mathematical expressions. Therefore, let’s delve into the step-by-step process of simplifying (-8x^3y)2 and explore the underlying concepts that make this simplification possible. This journey will equip you with the tools and knowledge to tackle similar mathematical challenges with confidence and precision.

To simplify the expression (-8x^3y)2, we will follow a step-by-step approach, carefully applying the rules of exponents. The first key rule we'll utilize is the power of a product rule, which dictates that when a product is raised to a power, each factor within the product is raised to that power individually. In our case, this means we need to apply the exponent 2 to each factor inside the parentheses: -8, x^3, and y. This step transforms the expression into (-8)^2 * (x³⁾2 * (y)^2. Now, let's break down each part of this expression. First, we have (-8)^2, which means -8 multiplied by itself: -8 * -8 = 64. Remember, a negative number multiplied by a negative number results in a positive number. Next, we address (x³⁾2. Here, we apply the power of a power rule, which states that when a power is raised to another power, we multiply the exponents. So, (x³⁾2 becomes x^(3*2) = x^6. This rule is crucial for simplifying expressions where exponents are nested. Finally, we have (y)^2, which is simply y squared, or y^2. There's no further simplification needed for this term. Putting it all together, we have 64 * x^6 * y^2. This is the simplified form of the original expression. This step-by-step approach highlights the importance of understanding and applying the rules of exponents correctly. It also demonstrates how breaking down a complex expression into smaller, manageable parts can make the simplification process much clearer and less daunting. By following this methodical approach, you can confidently simplify similar expressions and build a strong foundation in algebraic manipulation.

The power of a product rule is a cornerstone in simplifying expressions involving exponents. This rule, mathematically expressed as (ab)^n = a^n * b^n, states that when a product of terms is raised to a power, each term within the product must be raised to that power individually. This rule is not just a mathematical convenience; it stems from the fundamental definition of exponents as repeated multiplication. To illustrate this, let’s consider a simple example: (2x)^3. According to the power of a product rule, this should be equal to 2^3 * x^3. Let’s verify this. (2x)^3 means (2x) * (2x) * (2x). We can rearrange this multiplication as 2 * 2 * 2 * x * x * x, which is indeed 2^3 * x^3. This example provides a concrete understanding of why the rule works. Now, let’s apply this rule to our expression, (-8x^3y)2. In this expression, we have a product of three terms: -8, x^3, and y. The entire product is raised to the power of 2. According to the power of a product rule, we must raise each of these terms to the power of 2 individually. This gives us (-8)^2 * (x³⁾2 * (y)^2. This step is crucial because it breaks down the complex expression into simpler components that can be easily simplified. It's important to remember that the power of a product rule applies regardless of the number of terms in the product. Whether you have two terms, three terms, or more, each term must be raised to the power outside the parentheses. This rule is not only applicable in algebra but also in various other branches of mathematics and science, making it a fundamental concept to grasp. By understanding and mastering the power of a product rule, you gain a powerful tool for simplifying expressions and solving mathematical problems.

Another essential rule in simplifying expressions is the power of a power rule. This rule, expressed mathematically as (a^m)n = a^(mn), states that when a power is raised to another power, you multiply the exponents. This rule is particularly useful when dealing with nested exponents, as it allows us to consolidate them into a single exponent. To understand why this rule works, let’s consider an example: (x²⁾3. According to the power of a power rule, this should be equal to x^(23) = x^6. Let’s see why this is the case. (x²⁾3 means (x^2) * (x^2) * (x^2). Each x^2 term is x * x. So, we have (x * x) * (x * x) * (x * x), which is six x’s multiplied together, or x^6. This illustrates the underlying principle of the rule: raising a power to another power is equivalent to repeated multiplication of the base. Now, let’s apply this rule to the (x³⁾2 part of our expression. We have x^3 raised to the power of 2. According to the power of a power rule, we multiply the exponents: 3 * 2 = 6. Therefore, (x³⁾2 simplifies to x^6. This simplification is a direct application of the power of a power rule. It's crucial to remember that this rule applies only when a power is raised to another power. It does not apply when multiplying terms with the same base (in which case we add the exponents) or when adding or subtracting terms with exponents. The power of a power rule is a fundamental tool in simplifying expressions and solving equations involving exponents. It allows us to efficiently handle nested exponents and reduce complex expressions to their simplest forms. By mastering this rule, you can confidently tackle a wide range of mathematical problems involving exponents.

After applying the power of a product rule and the power of a power rule, we have simplified the expression (-8x^3y)2 to 64x^6y2. This final step involves putting all the simplified components together to arrive at the ultimate simplified form. We started with (-8x^3y)2, and we broke it down into (-8)^2 * (x³⁾2 * (y)^2. We then simplified each component individually. (-8)^2 became 64, as -8 multiplied by -8 equals 64. (x³⁾2 became x^6, using the power of a power rule, where we multiplied the exponents 3 and 2. (y)^2 remained as y^2, as it was already in its simplest form. Now, we combine these simplified components: 64 * x^6 * y^2. This is the simplified form of the original expression. It's important to note that this expression cannot be simplified further, as there are no like terms to combine and no more exponent rules to apply. The simplified expression 64x^6y2 represents the same value as the original expression (-8x^3y)2, but it is in a more concise and understandable form. This is the goal of simplification: to express a mathematical expression in its most basic form while preserving its value. This process of simplification is not just about finding the right answer; it's about understanding the underlying mathematical principles and applying them correctly. By following a step-by-step approach and carefully applying the rules of exponents, you can confidently simplify complex expressions and build a strong foundation in algebra. The ability to simplify expressions is a crucial skill in mathematics, and it opens the door to solving more advanced problems and understanding more complex concepts.

In conclusion, the simplification of the expression (-8x^3y)2 demonstrates the power and importance of understanding and applying the rules of exponents. By breaking down the expression into smaller parts and applying the power of a product rule and the power of a power rule, we were able to transform a seemingly complex expression into a simple and concise form: 64x^6y2. This process highlights the significance of a step-by-step approach in mathematics. Each step, from applying the power of a product rule to multiplying exponents using the power of a power rule, is crucial for arriving at the correct simplified expression. The ability to simplify expressions is not just a mathematical skill; it’s a problem-solving skill that can be applied in various contexts. Whether you're solving equations, graphing functions, or working with more advanced mathematical concepts, the ability to simplify expressions is essential for success. Furthermore, the principles learned in simplifying expressions extend beyond the realm of mathematics. The logical and methodical approach required for simplification can be applied to problem-solving in various fields, from science and engineering to finance and economics. By mastering the art of simplification, you develop a valuable tool for tackling complex problems and making informed decisions. Therefore, the journey of simplifying expressions is not just about finding the right answer; it’s about developing a mindset of clarity, precision, and problem-solving prowess. As you continue your mathematical journey, remember the principles and techniques discussed here, and you'll be well-equipped to tackle any challenges that come your way.