Simplifying Expressions A Step-by-Step Guide To Solving (√5+3√2)(√5-2√2)

Introduction

In this article, we will delve into the simplification of the expression $(\sqrt{5}+3 \sqrt{2})(\sqrt{5}-2 \sqrt{2})$. This type of problem often appears in mathematics, particularly in algebra and pre-calculus courses. Mastering the technique to simplify such expressions is crucial for students and anyone interested in enhancing their mathematical skills. We will break down the process step-by-step, ensuring a clear understanding of the underlying principles and methods. This introduction sets the stage for a comprehensive exploration of the simplification process, providing readers with the necessary context and motivation to dive deeper into the topic.

Understanding the Basics

Before we begin, it’s essential to understand the basic rules of algebraic manipulation and the properties of square roots. We will use the distributive property (also known as the FOIL method) to expand the expression. The distributive property states that for any numbers a, b, c, and d, $(a + b)(c + d) = ac + ad + bc + bd$. Additionally, we need to know how to multiply square roots: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. These fundamental concepts form the bedrock of our simplification process, enabling us to approach the problem with confidence and accuracy. Grasping these principles ensures that each step in the simplification is logical and sound, leading to the correct solution. Understanding these basics is not just about memorizing formulas; it’s about developing a solid foundation in mathematical thinking that can be applied to a wide range of problems.

Applying the Distributive Property

To simplify the expression $(\sqrt{5}+3 \sqrt{2})(\sqrt{5}-2 \sqrt{2})$, we apply the distributive property (FOIL method). This involves multiplying each term in the first parentheses with each term in the second parentheses. Here’s how it breaks down:

• First: $\sqrt{5} \cdot \sqrt{5} = 5$
• Outer: $\sqrt{5} \cdot (-2 \sqrt{2}) = -2 \sqrt{10}$
• Inner: $3 \sqrt{2} \cdot \sqrt{5} = 3 \sqrt{10}$
• Last: $3 \sqrt{2} \cdot (-2 \sqrt{2}) = -6 \cdot 2 = -12$

Each of these steps is crucial in expanding the expression, and accuracy at this stage is paramount. The distributive property ensures that we account for all possible products, setting the stage for subsequent simplification. By meticulously applying the FOIL method, we transform the original expression into a more manageable form that can be further simplified by combining like terms and simplifying square roots. This process highlights the power of algebraic manipulation in making complex expressions more understandable and solvable.

Step-by-Step Simplification

Expanding the Expression

Let's combine the results from the distributive property:

$(\sqrt{5}+3 \sqrt{2})(\sqrt{5}-2 \sqrt{2}) = 5 - 2 \sqrt{10} + 3 \sqrt{10} - 12$

This step brings together the individual products calculated earlier, creating a single, expanded expression. It is a critical juncture in the simplification process, where we transition from individual terms to a unified expression that can be further refined. Attention to detail is crucial here to ensure that all terms are correctly included and that their signs are accurately maintained. This expanded form serves as the foundation for the next steps, where we will combine like terms and simplify any remaining radicals.

Combining Like Terms

Now, we combine the like terms in the expression. We have the constants 5 and -12, and the terms involving $\sqrt{10}$.

$5 - 2 \sqrt{10} + 3 \sqrt{10} - 12 = (5 - 12) + (-2 \sqrt{10} + 3 \sqrt{10})$

This step involves identifying and grouping terms that share a common factor or radical. Combining like terms is a fundamental algebraic technique that simplifies expressions by reducing the number of terms and making the expression more concise. In this case, we combine the constant terms and the terms involving $\sqrt{10}$. This process not only simplifies the expression but also makes it easier to understand and work with in subsequent steps.

Simplifying Constants and Radicals

Performing the arithmetic, we get:

$(5 - 12) + (-2 \sqrt{10} + 3 \sqrt{10}) = -7 + \sqrt{10}$

Here, we’ve simplified the constants (5 - 12 = -7) and combined the terms involving $\sqrt{10}$ (-2 \sqrt{10} + 3 \sqrt{10} = \sqrt{10}$). This step represents the culmination of the simplification process, where we reduce the expression to its most basic form. By performing the arithmetic operations and combining the radical terms, we arrive at a simplified expression that is both concise and easy to interpret. This final simplified form is the solution we were seeking, and it demonstrates the power of algebraic manipulation in transforming complex expressions into simpler, more manageable forms.

Final Simplified Expression

Thus, the simplified expression is:

$\sqrt{10} - 7$

This is the final result of our simplification process. We have successfully transformed the original expression $(\sqrt{5}+3 \sqrt{2})(\sqrt{5}-2 \sqrt{2})$ into its simplest form, $\sqrt{10} - 7$. This final step underscores the elegance and efficiency of algebraic techniques in reducing complex expressions to their essential components. The result not only provides a solution to the problem but also showcases the power of systematic simplification in mathematics.

Conclusion

In summary, we simplified the expression $(\sqrt{5}+3 \sqrt{2})(\sqrt{5}-2 \sqrt{2})$ by applying the distributive property, combining like terms, and simplifying radicals. The final simplified form is $\sqrt{10} - 7$. This exercise demonstrates the importance of understanding and applying fundamental algebraic principles to solve mathematical problems. The ability to simplify expressions is a crucial skill in mathematics, enabling us to tackle more complex problems with confidence and precision. By mastering these techniques, students and enthusiasts alike can enhance their mathematical proficiency and develop a deeper appreciation for the elegance and power of algebraic manipulation.

Key Takeaways

• Distributive Property (FOIL): Essential for expanding expressions.
• Combining Like Terms: Simplifies expressions by grouping similar terms.
• Simplifying Radicals: Reduces radicals to their simplest form.

These key takeaways highlight the core techniques used in the simplification process. Each of these elements plays a crucial role in transforming complex expressions into simpler forms. The distributive property allows us to expand the expression, while combining like terms reduces the number of terms, and simplifying radicals ensures that the expression is in its most concise form. By mastering these techniques, individuals can confidently approach a wide range of algebraic simplification problems.

Practice Makes Perfect

To master the art of simplifying expressions, practice is crucial. Try simplifying similar expressions to reinforce your understanding and skills. The more you practice, the more comfortable and proficient you will become in applying these techniques. Practice not only solidifies your understanding but also helps you develop the intuition and problem-solving skills necessary to tackle more challenging mathematical problems. By consistently working through examples and exercises, you can build a strong foundation in algebra and enhance your overall mathematical abilities. Remember, each problem solved is a step towards mastery.

Answer

The correct answer is A. $\sqrt{10}-7$.

This confirms the result of our step-by-step simplification process. By carefully applying the distributive property, combining like terms, and simplifying radicals, we have arrived at the correct answer. This validation not only reinforces our understanding of the simplification process but also highlights the importance of accuracy and attention to detail in mathematical problem-solving. The correct answer serves as a testament to the effectiveness of the techniques we have employed and underscores the value of a systematic approach to simplifying algebraic expressions.