Multiply And Simplify 2w^7u^2 * 3w * 6u^5

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In mathematics, simplifying algebraic expressions is a fundamental skill. This involves combining like terms and applying the rules of exponents. In this article, we will delve into the process of multiplying algebraic expressions and simplifying the results. We will specifically address the problem of multiplying monomials, which are algebraic expressions consisting of a single term. Our focus will be on understanding the underlying principles and applying them to a given expression.

Understanding the Basics of Monomial Multiplication

To effectively multiply monomials, a solid understanding of the basic rules of exponents and the commutative and associative properties of multiplication is essential. Monomials, which are single-term algebraic expressions, often consist of coefficients (numerical factors) and variables raised to exponents. For example, consider the expression 2w⁷u² ⋅ 3w ⋅ 6u⁵. This expression involves the product of three monomials: 2w⁷u², 3w, and 6u⁵. To simplify this expression, we need to multiply the coefficients together and combine the variables with the same base by adding their exponents. This approach ensures that the final expression is in its simplest form, making it easier to work with in subsequent algebraic manipulations.

The commutative property allows us to change the order of factors without altering the product (e.g., a ⋅ b = b ⋅ a). The associative property allows us to regroup factors without changing the product (e.g., (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c)). These properties are crucial in rearranging and grouping like terms in an algebraic expression. In the given problem, we can rearrange the terms to group the coefficients together and the variables with the same base together. This makes the multiplication process more organized and less prone to errors. Understanding these fundamental properties lays the groundwork for more complex algebraic manipulations, such as factoring and solving equations.

Step-by-Step Guide to Multiplying Monomials

The process of multiplying monomials can be broken down into a series of clear steps. First, identify the coefficients in each monomial. These are the numerical factors that multiply the variables. For example, in the monomial 2w⁷u², the coefficient is 2. In the next step, multiply these coefficients together. This gives the numerical part of the simplified monomial. Next, identify the variables in each monomial. For each variable, add the exponents of the like variables. Remember that if a variable does not have an explicitly written exponent, it is understood to have an exponent of 1. For instance, in the expression 3w, the variable w has an exponent of 1. Finally, write the simplified monomial by combining the product of the coefficients with the variables raised to their combined exponents. This systematic approach ensures accuracy and efficiency in simplifying algebraic expressions.

For example, let’s consider multiplying x²y and x³y⁴. First, identify the coefficients: in this case, both monomials have a coefficient of 1 (since no number is explicitly written, it is understood to be 1). Multiply the coefficients: 1 * 1 = 1. Next, identify the variables and add their exponents: for x, we have x² and x³, so the combined exponent is 2 + 3 = 5. For y, we have y¹ and y⁴, so the combined exponent is 1 + 4 = 5. Finally, write the simplified monomial: 1 * x⁵ * y⁵, which is simply x⁵y⁵. This step-by-step process demystifies monomial multiplication and makes it accessible even for those new to algebra.

Applying the Rules to the Given Expression: 2w⁷u² ⋅ 3w ⋅ 6u⁵

Now, let’s apply the rules of monomial multiplication to the given expression: 2w⁷u² ⋅ 3w ⋅ 6u⁵. This example provides a practical application of the principles discussed earlier. The expression involves three monomials, each with coefficients and variables raised to exponents. Our goal is to simplify this expression by multiplying the coefficients and combining like variables. By breaking down the problem into manageable steps, we can arrive at the simplified form efficiently and accurately.

The first step is to identify and multiply the coefficients. In this expression, the coefficients are 2, 3, and 6. Multiplying these together, we get 2 * 3 * 6 = 36. This numerical value will be the coefficient of our simplified monomial. Next, we need to identify and combine the like variables. The variables in the expression are w and u. For the variable w, we have w⁷ and w¹ (remember that w without an exponent is understood to have an exponent of 1). Adding the exponents, we get 7 + 1 = 8, so we have w⁸. For the variable u, we have u² and u⁵. Adding the exponents, we get 2 + 5 = 7, so we have u⁷. Finally, we combine the product of the coefficients with the variables raised to their combined exponents. This gives us the simplified monomial 36w⁸u⁷. This example demonstrates how the step-by-step approach can effectively simplify complex algebraic expressions.

Detailed Breakdown of the Solution

To ensure a thorough understanding, let's break down the solution to the given expression step-by-step. This detailed breakdown will reinforce the principles discussed and provide clarity on each operation performed. Understanding each step is crucial for mastering the simplification of algebraic expressions. It also aids in troubleshooting errors and building confidence in your algebraic skills. By carefully examining each part of the process, we can ensure that the final answer is both accurate and easily understood.

  1. Identify the Coefficients: In the expression 2w⁷u² ⋅ 3w ⋅ 6u⁵, the coefficients are 2, 3, and 6. These are the numerical factors multiplying the variables. Identifying these coefficients is the first step in simplifying the expression. It allows us to separate the numerical part of the problem from the variable part, making the multiplication process more manageable. This initial separation is a common strategy in algebra, helping to break down complex problems into simpler components.
  2. Multiply the Coefficients: Multiply the coefficients together: 2 * 3 * 6 = 36. This gives us the numerical part of the simplified monomial. This step is a straightforward arithmetic operation, but it's essential for determining the final coefficient. The result, 36, will be the numerical factor in our simplified expression. Ensuring accuracy in this step is vital, as any error here will propagate through the rest of the solution.
  3. Identify Like Variables: The variables in the expression are w and u. We need to group like variables together to add their exponents. Recognizing like variables is a fundamental aspect of simplifying algebraic expressions. It allows us to apply the rules of exponents correctly. This step is crucial for combining terms and reducing the expression to its simplest form.
  4. Combine Exponents for w: For the variable w, we have w⁷ and w¹. Add the exponents: 7 + 1 = 8. This means the simplified expression will have w⁸. The rule for multiplying like variables states that we add their exponents. This is a core principle in algebra and is essential for simplifying expressions involving exponents. The result, w⁸, indicates that w is raised to the power of 8 in the simplified expression.
  5. Combine Exponents for u: For the variable u, we have u² and u⁵. Add the exponents: 2 + 5 = 7. This means the simplified expression will have u⁷. Similar to the previous step, we apply the rule for multiplying like variables by adding their exponents. The result, u⁷, shows that u is raised to the power of 7 in the simplified expression.
  6. Write the Simplified Monomial: Combine the product of the coefficients (36) with the variables raised to their combined exponents (w⁸ and u⁷). The simplified monomial is 36w⁸u⁷. This final step brings together all the previous steps, combining the numerical coefficient and the variable terms. The result, 36w⁸u⁷, is the simplified form of the original expression. It represents the most concise way to express the product of the given monomials.

Common Mistakes to Avoid

When multiplying and simplifying algebraic expressions, several common mistakes can occur. Recognizing and avoiding these errors is crucial for achieving accurate results. One common mistake is forgetting to add the exponents correctly when multiplying variables with the same base. For example, when multiplying w⁷ and w, some might incorrectly write w⁷ instead of w⁸. Another frequent error is incorrectly multiplying coefficients. A simple arithmetic mistake can lead to a completely wrong answer. For instance, in the given problem, if the coefficients 2, 3, and 6 are multiplied incorrectly, the entire solution will be flawed. Additionally, students sometimes mix up the rules for adding and multiplying exponents. It's essential to remember that when multiplying like variables, we add the exponents, but when raising a power to a power, we multiply the exponents. Being mindful of these common pitfalls can significantly improve your accuracy in simplifying algebraic expressions.

Another mistake is not simplifying the expression completely. For example, if after multiplying the coefficients and combining like variables, the expression still has terms that can be further simplified, the solution is not complete. It's important to always check the final result to ensure it is in its simplest form. Furthermore, students might forget to include variables with an exponent of 1. Remember, if a variable appears without an explicit exponent, it is understood to have an exponent of 1. Overlooking this can lead to incorrect exponent addition. Finally, a lack of attention to detail can result in errors. Always double-check each step, from identifying the coefficients to combining the variables, to minimize mistakes. Careful and methodical work is key to success in algebra.

Practice Problems

To solidify your understanding of multiplying and simplifying algebraic expressions, practice is essential. Working through various problems helps reinforce the rules and techniques discussed. Here are some practice problems to try:

  1. Simplify: 4x³y² ⋅ 5xy⁴
  2. Multiply and simplify: (2a²b)(3ab³)(4a³)
  3. Simplify: (7p⁴q)(2p²q⁵)(3q²)
  4. Multiply: 9m⁵n ⋅ 2mn³ ⋅ 5m²
  5. Simplify: (6c³d²)(c⁴d)(2d³)

Working through these problems will provide valuable experience in applying the rules of exponents and combining like terms. Be sure to break down each problem into steps, as demonstrated earlier, to ensure accuracy. After solving each problem, check your answers to confirm your understanding. If you encounter any difficulties, revisit the explanations and examples provided in this article. Consistent practice is the key to mastering algebraic simplification.

Solutions to Practice Problems

To help you check your work, here are the solutions to the practice problems:

  1. 4x³y² ⋅ 5xy⁴ = 20x⁴y⁶
  2. (2a²b)(3ab³)(4a³) = 24a⁶b⁴
  3. (7p⁴q)(2p²q⁵)(3q²) = 42p⁶q⁸
  4. 9m⁵n ⋅ 2mn³ ⋅ 5m² = 90m⁸n⁴
  5. (6c³d²)(c⁴d)(2d³) = 12c⁷d⁶

Review these solutions carefully. If your answers match, you have a good grasp of the concepts. If you made any mistakes, identify the specific step where the error occurred and review the relevant section of this article. Understanding the mistakes and correcting them is a crucial part of the learning process. Don't hesitate to try similar problems to reinforce your skills and build confidence in your ability to simplify algebraic expressions.

Conclusion

Multiplying and simplifying algebraic expressions is a fundamental skill in algebra. By understanding the rules of exponents, the commutative and associative properties, and following a step-by-step approach, you can effectively simplify complex expressions. Remember to pay attention to detail, avoid common mistakes, and practice regularly to master this skill. The principles and techniques discussed in this article provide a solid foundation for more advanced algebraic concepts. With consistent effort and practice, you can become proficient in simplifying algebraic expressions, which will be invaluable in your mathematical journey.

By mastering these skills, you'll be well-equipped to tackle more complex algebraic problems. The ability to simplify expressions efficiently and accurately is essential for success in higher-level mathematics courses and various applications in science and engineering. Embrace the challenge of algebra, and continue to build your skills through practice and perseverance. The rewards of a strong algebraic foundation are substantial, opening doors to a deeper understanding of mathematics and its applications in the world around us.