How Do You Find The Product Of The Expression (7x²y³)(3x⁵y⁸)?

In the realm of mathematics, particularly in algebra, understanding how to simplify expressions is a fundamental skill. This article will delve into the process of finding the product of algebraic expressions, using the specific example of (7x²y³)(3x⁵y⁸). We'll break down each step, ensuring a clear understanding of the underlying principles and arrive at the correct solution. This knowledge is essential for tackling more complex algebraic problems and for building a solid foundation in mathematics.

Deconstructing the Expression: A Step-by-Step Guide

When faced with an expression like (7x²y³)(3x⁵y⁸), the initial approach involves identifying the different components and understanding how they interact. The expression consists of two terms, each a product of a coefficient (a numerical factor) and variables raised to certain powers. The first term is 7x²y³, where 7 is the coefficient, x is a variable raised to the power of 2, and y is a variable raised to the power of 3. The second term is 3x⁵y⁸, where 3 is the coefficient, x is a variable raised to the power of 5, and y is a variable raised to the power of 8.

To find the product of these two terms, we need to apply the rules of exponents and multiplication. This involves multiplying the coefficients together and then multiplying the variables with the same base by adding their exponents. The core principle here is the product of powers rule, which states that when multiplying exponents with the same base, you add the powers. This rule is the cornerstone of simplifying this type of expression and understanding it thoroughly is crucial for success.

Therefore, the first step in simplifying (7x²y³)(3x⁵y⁸) is to separate the coefficients and variables, setting the stage for applying the multiplication and exponent rules. This meticulous approach ensures that each component is handled correctly, leading to the accurate simplification of the expression. This foundational understanding is paramount for more complex algebraic manipulations and problem-solving scenarios.

Multiplying the Coefficients: The Numerical Foundation

The first concrete step in simplifying the expression (7x²y³)(3x⁵y⁸) is to multiply the coefficients. Coefficients are the numerical parts of the terms, in this case, 7 and 3. Multiplying these two numbers together is a straightforward arithmetic operation: 7 multiplied by 3 equals 21. This numerical product forms the foundation of our simplified expression, setting the stage for combining the variable components. Understanding this basic multiplication is essential as it directly impacts the final result and underscores the importance of numerical accuracy in algebraic simplification.

The result, 21, becomes the coefficient of the simplified term. It represents the numerical factor that will be multiplied by the variable terms. This step highlights the importance of basic arithmetic skills in algebra, as an error in multiplying the coefficients will propagate through the rest of the simplification process. The careful and accurate multiplication of coefficients is thus a crucial step in achieving the correct final answer.

Furthermore, this process underscores a fundamental principle in algebra: treating numerical and variable components separately before combining them. This approach allows for a systematic and organized simplification process, reducing the chances of errors and ensuring a clear understanding of each step. Mastering this separation and multiplication of coefficients is a key building block for more advanced algebraic manipulations.

Combining the Variables: Applying the Rules of Exponents

After successfully multiplying the coefficients, the next crucial step in simplifying (7x²y³)(3x⁵y⁸) involves combining the variables. This requires a solid understanding of the rules of exponents, particularly the product of powers rule. This rule states that when multiplying variables with the same base, you add their exponents. In our expression, we have two variables, x and y, each raised to different powers.

For the variable x, we have x² in the first term and x⁵ in the second term. According to the product of powers rule, we add the exponents: 2 + 5 = 7. This means that the x term in the simplified expression will be x⁷. Similarly, for the variable y, we have y³ in the first term and y⁸ in the second term. Adding the exponents, we get 3 + 8 = 11. Therefore, the y term in the simplified expression will be y¹¹.

The product of powers rule is a cornerstone of algebraic simplification. It allows us to efficiently combine variables and express them in their simplest form. Understanding and applying this rule correctly is essential for solving a wide range of algebraic problems. This step demonstrates how algebraic rules provide a structured way to manipulate expressions and arrive at a concise and accurate result. This is an important point to emphasize.

By carefully applying the product of powers rule to both x and y, we have successfully combined the variable components of the expression. This step, combined with the earlier multiplication of coefficients, brings us closer to the final simplified form of the expression. This systematic approach, breaking down the problem into manageable steps, is a key strategy for success in algebra.

The Final Product: Putting It All Together

Having meticulously multiplied the coefficients and combined the variables using the product of powers rule, we are now ready to present the final simplified product of (7x²y³)(3x⁵y⁸). Recall that the product of the coefficients, 7 and 3, is 21. The combined x term is x⁷ (obtained by adding the exponents 2 and 5), and the combined y term is y¹¹ (obtained by adding the exponents 3 and 8).

Therefore, the final simplified expression is 21x⁷y¹¹. This represents the product of the two original terms in its most concise form. The expression clearly shows the coefficient and the variables, each raised to their respective powers. This result is a direct application of the fundamental rules of algebra, demonstrating the power of these rules in simplifying complex expressions.

This final product, 21x⁷y¹¹, is not just an answer; it is a testament to the step-by-step approach we have taken. By breaking down the problem into smaller, manageable parts, we have ensured accuracy and clarity. This approach is crucial for tackling more complex algebraic problems and for building a strong foundation in mathematics. The journey from the initial expression to the final product highlights the elegance and efficiency of algebraic principles. This is the most important takeaway.

Identifying the Correct Answer: Multiple Choice Analysis

Now that we have derived the simplified product, 21x⁷y¹¹, let's analyze the multiple-choice options provided in the original question. The options are:

A. 10x⁷y¹¹ B. 10x¹⁰y²⁴ C. 21x⁷y¹¹ D. 21x¹⁰y²⁴

By comparing our derived answer, 21x⁷y¹¹, with the options, it becomes clear that option C, 21x⁷y¹¹, is the correct answer. This confirms the accuracy of our step-by-step simplification process. Options A, B, and D are incorrect because they either have the wrong coefficient or incorrect exponents for the variables.

This multiple-choice analysis highlights the importance of carefully reviewing the options and comparing them with the derived solution. It also underscores the value of a systematic approach in solving algebraic problems, as it allows for a clear and confident identification of the correct answer. This process of verification is a crucial skill in mathematics, ensuring accuracy and minimizing errors. Remember to always double-check your results.

Conclusion: Mastering Algebraic Products

In conclusion, this article has provided a comprehensive guide to finding the product of algebraic expressions, using the example of (7x²y³)(3x⁵y⁸). We have explored the step-by-step process, starting with deconstructing the expression, multiplying the coefficients, combining the variables using the product of powers rule, and finally arriving at the simplified product, 21x⁷y¹¹.

This journey through algebraic simplification highlights the importance of understanding the fundamental rules and principles. The product of powers rule, in particular, plays a crucial role in combining variables and simplifying expressions. By mastering these concepts, you can confidently tackle a wide range of algebraic problems. This article also emphasized the value of a systematic approach, breaking down complex problems into smaller, manageable steps, ensuring accuracy and clarity.

The ability to simplify algebraic expressions is a fundamental skill in mathematics, serving as a building block for more advanced topics. By understanding the underlying principles and practicing regularly, you can develop a strong foundation in algebra and excel in your mathematical pursuits. This knowledge is not just for academic success; it is a valuable tool for problem-solving in various real-world scenarios. Therefore, mastering algebraic products is an investment in your mathematical future and beyond.

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