Simplifying 2x * 5w^9 * 2x^3w^3 A Step-by-Step Guide

Introduction

In this article, we will delve into the process of multiplying algebraic expressions, with a particular focus on simplifying expressions involving exponents. The given expression we will be working with is 2x ⋅ 5w⁹ ⋅ 2x³w³. Our goal is to simplify this expression as much as possible by applying the rules of exponents and combining like terms. This is a fundamental concept in algebra, and mastering it is crucial for solving more complex mathematical problems. Understanding how to manipulate expressions with exponents not only helps in simplifying algebraic equations but also lays the groundwork for calculus and other advanced mathematical topics. The ability to simplify expressions efficiently can also save time and reduce errors in mathematical calculations, making it an invaluable skill for students and professionals alike. In this guide, we will break down the process step by step, providing clear explanations and examples to ensure a thorough understanding. This approach will help you not only solve this specific problem but also equip you with the knowledge to tackle similar problems with confidence. By the end of this article, you should be able to confidently multiply and simplify various algebraic expressions, enhancing your overall mathematical proficiency. Remember, practice is key to mastering these concepts, so we encourage you to try out similar problems on your own after going through this guide. This hands-on experience will solidify your understanding and improve your problem-solving skills.

Understanding the Basics of Exponents

Before we dive into simplifying the expression, it’s important to understand the basic rules of exponents. An exponent indicates how many times a base number is multiplied by itself. For instance, w⁹ means w multiplied by itself nine times (w * w * w * w * w * w * w * w * w). When multiplying terms with the same base, we add their exponents. This rule is crucial for simplifying expressions like the one we are working with. The general rule can be expressed as aᵐ * aⁿ = aᵐ⁺ⁿ, where 'a' is the base and 'm' and 'n' are the exponents. This rule applies because when you multiply, you're essentially combining the repeated multiplications. For example, if you have x² * x³, it means (x * x) * (x * x * x), which is x⁵. Understanding this fundamental principle makes simplifying expressions much more straightforward. Moreover, it's important to recognize that this rule only applies when the bases are the same. You cannot directly add exponents if the bases are different, such as in the case of x² * y³. In such cases, the terms remain separate unless there are other operations that can be performed. Grasping these basics will help you navigate through the simplification process more effectively and avoid common mistakes. Additionally, familiarity with these rules makes more advanced algebraic manipulations easier to handle, providing a solid foundation for further mathematical studies. This section aims to ensure that you have a clear understanding of the underlying principles, setting you up for success in simplifying the given expression and similar problems.

Step-by-Step Simplification of the Expression

Let's break down the simplification of the expression 2x ⋅ 5w⁹ ⋅ 2x³w³ step by step. First, we can rearrange the terms to group the coefficients (the numerical parts) and the variables with the same base together. This rearrangement doesn't change the value of the expression due to the commutative property of multiplication, which states that the order in which numbers are multiplied does not affect the result. So, we can rewrite the expression as 2 ⋅ 5 ⋅ 2 ⋅ x ⋅ x³ ⋅ w⁹ ⋅ w³. This regrouping makes it easier to identify like terms and apply the rules of exponents. Next, we multiply the coefficients together: 2 * 5 * 2 = 20. This simplifies the numerical part of the expression. Now, let's focus on the variables with exponents. We have x ⋅ x³. Remember that x is the same as x¹, so we can apply the rule of exponents which states aᵐ * aⁿ = aᵐ⁺ⁿ. Therefore, x¹ * x³ = x¹⁺³ = x⁴. Similarly, for the w terms, we have w⁹ ⋅ w³. Applying the same rule, we get w⁹⁺³ = w¹². Now we combine all the simplified parts together. We have the coefficient 20, the x term x⁴, and the w term w¹². Putting these together, we get the simplified expression: 20x⁴w¹². This is the final simplified form of the given expression. Each step in this process is crucial to ensure accuracy. By breaking down the problem into smaller, manageable parts, we can systematically simplify the expression and arrive at the correct answer. Understanding this step-by-step approach is vital for tackling more complex algebraic expressions in the future.

Applying the Rules of Exponents

To further illustrate how the rules of exponents are applied in simplifying expressions, let's focus on the core concept: aᵐ * aⁿ = aᵐ⁺ⁿ. This rule is fundamental when multiplying terms with the same base. In our expression, 2x ⋅ 5w⁹ ⋅ 2x³w³, we encountered terms with the bases x and w. When multiplying x (which is x¹) by x³, we add the exponents: 1 + 3 = 4, resulting in x⁴. Similarly, when multiplying w⁹ by w³, we add the exponents: 9 + 3 = 12, resulting in w¹². This rule works because exponents represent repeated multiplication. When you multiply terms with the same base, you're essentially combining these repeated multiplications. For example, x³ means x * x * x, and x¹ means x. Multiplying them together, you get x * x * x * x, which is x⁴. Understanding this rationale helps in remembering and applying the rule correctly. It's important to note that this rule only applies when the bases are the same. You cannot add the exponents of terms with different bases directly. For instance, x⁴ and w¹² cannot be combined further because they have different bases. Moreover, it's crucial to identify the base and the exponent correctly. The base is the number or variable being raised to a power, and the exponent is the power to which the base is raised. Misidentification can lead to incorrect simplification. By consistently applying this rule and understanding its underlying principle, you can confidently simplify a wide range of algebraic expressions. This skill is essential for success in algebra and higher-level mathematics, as it forms the basis for more complex operations and problem-solving techniques. Regular practice and careful attention to detail are key to mastering this concept.

Combining Like Terms in the Expression

Combining like terms is a critical step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. In the expression 2x ⋅ 5w⁹ ⋅ 2x³w³, we identified the terms with the same variables (x and w) and grouped them together. This process is essential because it allows us to apply the rules of exponents effectively. Before combining, it’s helpful to rearrange the expression as we did earlier: 2 ⋅ 5 ⋅ 2 ⋅ x ⋅ x³ ⋅ w⁹ ⋅ w³. This rearrangement makes it easier to see which terms can be combined. We combined x and x³ because they both have the same base (x). Similarly, we combined w⁹ and w³ because they both have the same base (w). The coefficients (2, 5, and 2) are multiplied together separately since they are numerical constants and don't have any variables. It's important to note that terms with different variables or the same variable raised to different powers are not considered like terms and cannot be combined directly. For example, x⁴ and w¹² are not like terms and remain separate in the simplified expression. The process of combining like terms simplifies the expression by reducing the number of terms and making it easier to work with. It's a fundamental skill in algebra and is used extensively in solving equations, simplifying formulas, and performing other algebraic manipulations. By mastering this skill, you can tackle more complex problems with greater confidence and accuracy. Pay close attention to the variables and their exponents when combining like terms, and always double-check your work to ensure that you have combined the correct terms. This practice will help you avoid common errors and improve your overall algebraic proficiency.

Final Simplified Answer: 20x⁴w¹²

After carefully following the steps of rearranging terms, multiplying coefficients, and applying the rules of exponents, we have arrived at the final simplified answer for the expression 2x ⋅ 5w⁹ ⋅ 2x³w³, which is 20x⁴w¹². This result represents the most simplified form of the original expression, where all like terms have been combined and the expression is written in its most compact form. The coefficient 20 is the product of the numerical constants (2 * 5 * 2), and the variables x and w are raised to the appropriate powers after applying the rule aᵐ * aⁿ = aᵐ⁺ⁿ. The x⁴ term results from combining x¹ and x³, while the w¹² term results from combining w⁹ and w³. This final answer demonstrates the power of algebraic simplification, where a seemingly complex expression can be reduced to a much simpler form. Understanding how to arrive at this answer involves a solid grasp of the rules of exponents, the commutative property of multiplication, and the process of combining like terms. It's important to double-check your work to ensure that no further simplification is possible and that the answer is expressed in its simplest form. The ability to simplify expressions like this is a fundamental skill in algebra and is essential for success in more advanced mathematical topics. It not only simplifies calculations but also provides a deeper understanding of the relationships between variables and constants. By practicing and mastering these techniques, you can confidently tackle a wide range of algebraic problems and enhance your overall mathematical proficiency.

Conclusion

In conclusion, simplifying the expression 2x ⋅ 5w⁹ ⋅ 2x³w³ to 20x⁴w¹² involved a series of essential algebraic steps. We began by understanding the basic rules of exponents, particularly the rule for multiplying terms with the same base: aᵐ * aⁿ = aᵐ⁺ⁿ. We then applied this rule by rearranging the terms in the expression to group like terms together, making it easier to combine them. The coefficients were multiplied, and the exponents of the same bases were added. This systematic approach not only simplifies the expression but also reinforces the fundamental principles of algebra. Mastering these techniques is crucial for success in mathematics, as they form the basis for more complex algebraic manipulations and problem-solving. The ability to simplify expressions efficiently saves time, reduces errors, and provides a deeper understanding of mathematical concepts. Throughout this article, we have emphasized the importance of breaking down complex problems into smaller, manageable steps. This approach is valuable not only in algebra but also in various other areas of mathematics and problem-solving. By consistently applying these methods and practicing regularly, you can improve your algebraic skills and gain confidence in your ability to tackle a wide range of mathematical challenges. Remember, the key to mastering mathematics is practice and a solid understanding of the fundamental principles. This guide has provided you with the tools and knowledge to simplify expressions effectively, and we encourage you to continue practicing and exploring the world of algebra. With dedication and effort, you can achieve proficiency and excel in your mathematical journey.