Simplify (√2+√3)(√5-√7) A Comprehensive Solution

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In the realm of mathematics, simplifying expressions is a fundamental skill that allows us to manipulate and understand complex equations and formulas. This article delves into the process of simplifying a specific expression involving square roots: $(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7})$. We will walk through the steps involved in expanding and simplifying this expression, highlighting key mathematical principles and techniques along the way.

Understanding the Expression

The given expression, $(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7})$, involves the product of two binomials, each containing square roots. To simplify this expression, we need to apply the distributive property, also known as the FOIL (First, Outer, Inner, Last) method. This method ensures that each term in the first binomial is multiplied by each term in the second binomial.

Before we begin, let's define some key terms:

• Binomial: An algebraic expression consisting of two terms.
• Square Root: A number that, when multiplied by itself, equals a given number.
• Distributive Property: A property that allows us to multiply a sum by multiplying each addend separately and then adding the products.
• FOIL Method: A mnemonic for the distributive property, where FOIL stands for First, Outer, Inner, Last.

Applying the Distributive Property (FOIL Method)

To simplify the expression $(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7})$, we will use the FOIL method. This involves multiplying the First, Outer, Inner, and Last terms of the two binomials.

1. First: Multiply the first terms of each binomial: $\sqrt{2} * \sqrt{5} = \sqrt{10}$
2. Outer: Multiply the outer terms of the binomials: $\sqrt{2} * (-\sqrt{7}) = -\sqrt{14}$
3. Inner: Multiply the inner terms of the binomials: $\sqrt{3} * \sqrt{5} = \sqrt{15}$
4. Last: Multiply the last terms of each binomial: $\sqrt{3} * (-\sqrt{7}) = -\sqrt{21}$

Now, we add these products together:

$\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}$

Simplifying the Result

The expression we obtained, $\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}$, consists of four terms, each involving the square root of a different number. To determine if we can further simplify this expression, we need to check if any of the square roots can be simplified. A square root can be simplified if the number under the radical (the radicand) has any perfect square factors other than 1.

Let's examine each radicand:

• 10 = 2 * 5 (No perfect square factors)
• 14 = 2 * 7 (No perfect square factors)
• 15 = 3 * 5 (No perfect square factors)
• 21 = 3 * 7 (No perfect square factors)

Since none of the radicands have perfect square factors, the square roots cannot be simplified further. Additionally, none of the terms are like terms, meaning they cannot be combined. Therefore, the simplified form of the expression is:

$\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}$

Conclusion

In this article, we have successfully simplified the expression $(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7})$ by applying the distributive property (FOIL method) and simplifying the resulting terms. The simplified expression is $\sqrt{10} - \sqrt{14} + \sqrt{15} - \sqrt{21}$. This process demonstrates the importance of understanding and applying fundamental algebraic principles to manipulate and simplify mathematical expressions. Mastering these techniques is crucial for success in various areas of mathematics and related fields. By breaking down the problem into smaller steps and carefully applying the appropriate rules, we can confidently tackle complex expressions and arrive at accurate solutions.

Simplifying expressions involving square roots is a fundamental skill in mathematics, especially in algebra and calculus. The ability to manipulate and reduce these expressions is crucial for solving equations, evaluating functions, and understanding more complex mathematical concepts. In this comprehensive guide, we will delve deeper into the techniques and strategies for simplifying expressions with square roots, providing numerous examples and explanations to ensure a thorough understanding.

Core Principles of Simplifying Square Roots

Before we dive into specific examples, let's review the core principles that govern the simplification of square roots. These principles are based on the properties of radicals and the laws of exponents.

1. Product Property of Square Roots

The product property states that the square root of a product is equal to the product of the square roots. Mathematically, this is expressed as:

$\sqrt{ab} = \sqrt{a} * \sqrt{b}$

This property is essential for breaking down a square root into simpler factors. For example, $\sqrt{32}$ can be simplified by recognizing that 32 = 16 * 2, where 16 is a perfect square:

$\sqrt{32} = \sqrt{16 * 2} = \sqrt{16} * \sqrt{2} = 4\sqrt{2}$

2. Quotient Property of Square Roots

The quotient property states that the square root of a quotient is equal to the quotient of the square roots. Mathematically, this is expressed as:

$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

This property is useful for simplifying square roots of fractions. For example, $\sqrt{\frac{25}{9}}$ can be simplified as:

$\sqrt{\frac{25}{9}} = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3}$

3. Simplifying Radicals by Factoring

The most common technique for simplifying square roots involves factoring the radicand (the number under the square root) to identify perfect square factors. A perfect square is a number that can be expressed as the square of an integer (e.g., 4, 9, 16, 25). When a perfect square factor is identified, its square root can be taken, simplifying the overall expression.

For instance, let's simplify $\sqrt{75}$. First, we factor 75 into its prime factors:

75 = 3 * 25 = 3 * 5^2

We see that 25 is a perfect square (5^2). Therefore, we can rewrite the square root as:

$\sqrt{75} = \sqrt{3 * 5^2} = \sqrt{5^2} * \sqrt{3} = 5\sqrt{3}$

4. Combining Like Radicals

Like radicals are terms that have the same radicand. These terms can be combined by adding or subtracting their coefficients. For example:

$3\sqrt{2} + 5\sqrt{2} = (3+5)\sqrt{2} = 8\sqrt{2}$

However, terms with different radicands cannot be combined directly. For example, $3\sqrt{2}$ and $5\sqrt{3}$ cannot be combined because the radicands (2 and 3) are different.

Step-by-Step Simplification Process

To effectively simplify square root expressions, follow these steps:

1. Factor the Radicand: Identify the prime factors of the number under the square root.
2. Identify Perfect Square Factors: Look for factors that are perfect squares (e.g., 4, 9, 16, 25).
3. Apply the Product Property: Separate the perfect square factors from the remaining factors using the product property of square roots.
4. Take the Square Root of Perfect Squares: Simplify the square roots of the perfect square factors.
5. Combine Like Radicals (if applicable): If there are multiple terms with the same radicand, combine them by adding or subtracting their coefficients.
6. Simplify Fractions (if applicable): Use the quotient property to simplify square roots of fractions.

Examples of Simplifying Square Root Expressions

Let's walk through several examples to illustrate the simplification process.

Example 1: Simplify $\sqrt{48}$

1. Factor the Radicand: 48 = 2 * 24 = 2 * 2 * 12 = 2 * 2 * 2 * 6 = 2 * 2 * 2 * 2 * 3 = 2^4 * 3
2. Identify Perfect Square Factors: 2^4 is a perfect square (16)
3. Apply the Product Property: $\sqrt{48} = \sqrt{2^4 * 3} = \sqrt{2^4} * \sqrt{3}$
4. Take the Square Root of Perfect Squares: $\sqrt{2^4} = 2^2 = 4$
5. Simplified Expression: $\sqrt{48} = 4\sqrt{3}$

Example 2: Simplify $\sqrt{180}$

1. Factor the Radicand: 180 = 2 * 90 = 2 * 2 * 45 = 2 * 2 * 3 * 15 = 2 * 2 * 3 * 3 * 5 = 2^2 * 3^2 * 5
2. Identify Perfect Square Factors: 2^2 and 3^2 are perfect squares
3. Apply the Product Property: $\sqrt{180} = \sqrt{2^2 * 3^2 * 5} = \sqrt{2^2} * \sqrt{3^2} * \sqrt{5}$
4. Take the Square Root of Perfect Squares: $\sqrt{2^2} = 2$, $\sqrt{3^2} = 3$
5. Simplified Expression: $\sqrt{180} = 2 * 3 * \sqrt{5} = 6\sqrt{5}$

Example 3: Simplify $\frac{\sqrt{50}}{\sqrt{2}}$

1. Apply the Quotient Property: $\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}}$
2. Simplify the Fraction: $\frac{50}{2} = 25$
3. Take the Square Root: $\sqrt{25} = 5$
4. Simplified Expression: $\frac{\sqrt{50}}{\sqrt{2}} = 5$

Example 4: Simplify $3\sqrt{8} + 2\sqrt{18} - \sqrt{32}$

1. Simplify Each Square Root:
   • $\sqrt{8} = \sqrt{2^2 * 2} = 2\sqrt{2}$
   • $\sqrt{18} = \sqrt{3^2 * 2} = 3\sqrt{2}$
   • $\sqrt{32} = \sqrt{2^4 * 2} = 4\sqrt{2}$
2. Substitute Simplified Radicals: $3(2\sqrt{2}) + 2(3\sqrt{2}) - 4\sqrt{2}$
3. Multiply Coefficients: $6\sqrt{2} + 6\sqrt{2} - 4\sqrt{2}$
4. Combine Like Radicals: $(6 + 6 - 4)\sqrt{2} = 8\sqrt{2}$
5. Simplified Expression: $3\sqrt{8} + 2\sqrt{18} - \sqrt{32} = 8\sqrt{2}$

Advanced Techniques and Considerations

Rationalizing the Denominator

In some cases, square root expressions appear in the denominator of a fraction. To simplify these expressions, it's common practice to rationalize the denominator. Rationalizing the denominator involves eliminating the square root from the denominator by multiplying both the numerator and denominator by a suitable expression.

For example, to rationalize the denominator of $\frac{1}{\sqrt{2}}$, we multiply both the numerator and denominator by $\sqrt{2}$:

$\frac{1}{\sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$

Complex Expressions

Simplifying more complex expressions may involve multiple steps and the application of various algebraic techniques, such as the distributive property, FOIL method, and factoring. It's essential to break down the expression into smaller parts and apply the appropriate simplification rules to each part.

Conclusion

Simplifying expressions with square roots is a crucial skill in mathematics that requires a solid understanding of the core principles and techniques. By mastering these methods, you can effectively manipulate and reduce square root expressions, making them easier to work with in various mathematical contexts. Remember to practice regularly and break down complex problems into simpler steps to ensure accuracy and efficiency. With time and effort, you'll become proficient in simplifying even the most challenging square root expressions.

Solving mathematical problems is a cornerstone of mathematical education and a critical skill in various fields, including science, engineering, economics, and computer science. The ability to approach problems systematically, apply relevant concepts, and arrive at accurate solutions is highly valued. This article provides a detailed guide to effective problem-solving strategies, mathematical principles, and techniques, ensuring a comprehensive understanding and mastery of mathematical problem-solving.

Understanding the Problem-Solving Process

Effective problem-solving involves more than just applying formulas and procedures. It requires a structured approach that encompasses several key steps. Understanding and mastering this process can significantly improve your problem-solving abilities.

1. Understand the Problem

The first and most crucial step is to thoroughly understand the problem. This involves reading the problem carefully, identifying the key information, and clarifying any ambiguities. Ask yourself the following questions:

• What is the problem asking?
• What information is given?
• What are the knowns and unknowns?
• Are there any constraints or conditions?

2. Devise a Plan

Once you understand the problem, the next step is to develop a plan for solving it. This involves identifying the relevant mathematical concepts, formulas, and techniques that can be applied. Consider different approaches and choose the most efficient one.

Common problem-solving strategies include:

• Trial and Error: Try different approaches and see what works.
• Work Backwards: Start from the desired outcome and work backwards to find the initial conditions.
• Look for a Pattern: Identify patterns and use them to generalize a solution.
• Draw a Diagram: Visualize the problem using diagrams or graphs.
• Simplify the Problem: Break the problem into smaller, more manageable parts.
• Use a Formula or Theorem: Apply relevant mathematical formulas and theorems.

3. Carry Out the Plan

With a plan in place, the next step is to implement it. This involves performing the necessary calculations, applying the chosen techniques, and working through the steps in a systematic manner. Accuracy and attention to detail are crucial at this stage.

4. Look Back

After obtaining a solution, the final step is to look back and check your work. This involves verifying that the solution is reasonable, accurate, and satisfies the conditions of the problem. Consider alternative solutions and reflect on the problem-solving process.

Ask yourself the following questions:

• Is the solution correct?
• Does the solution make sense in the context of the problem?
• Could the problem have been solved in a different way?
• What did I learn from solving this problem?

Mathematical Principles and Techniques

To solve mathematical problems effectively, it's essential to have a solid understanding of mathematical principles and techniques. Here are some key areas to focus on:

1. Algebra

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating them. Key algebraic concepts include:

• Variables and Expressions: Understand how to represent unknown quantities using variables and how to write and simplify algebraic expressions.
• Equations and Inequalities: Learn how to solve linear, quadratic, and other types of equations and inequalities.
• Systems of Equations: Solve systems of linear and non-linear equations using methods such as substitution, elimination, and matrices.
• Factoring: Master factoring techniques for polynomials.
• Functions: Understand the concept of a function and learn how to graph and analyze different types of functions (linear, quadratic, exponential, logarithmic, etc.).

2. Geometry

Geometry deals with the properties and relationships of points, lines, angles, surfaces, and solids. Key geometric concepts include:

• Basic Geometric Figures: Understand the properties of triangles, quadrilaterals, circles, and other geometric figures.
• Area and Volume: Learn how to calculate the area of two-dimensional figures and the volume of three-dimensional solids.
• Congruence and Similarity: Understand the concepts of congruence and similarity and apply them to solve geometric problems.
• Trigonometry: Study trigonometric ratios and functions and their applications in solving triangles and other geometric problems.
• Coordinate Geometry: Use coordinate systems to represent geometric figures and solve problems using algebraic techniques.

3. Calculus

Calculus is a branch of mathematics that deals with continuous change. Key calculus concepts include:

• Limits and Continuity: Understand the concepts of limits and continuity and their role in calculus.
• Derivatives: Learn how to calculate derivatives and use them to find rates of change, slopes of tangent lines, and extrema of functions.
• Integrals: Understand the concept of integration and learn how to calculate definite and indefinite integrals.
• Applications of Calculus: Apply calculus to solve problems in physics, engineering, economics, and other fields.

4. Discrete Mathematics

Discrete mathematics deals with mathematical structures that are fundamentally discrete rather than continuous. Key discrete mathematics concepts include:

• Set Theory: Understand the basic concepts of set theory and operations on sets.
• Logic: Learn about propositional and predicate logic and their applications in mathematical reasoning.
• Combinatorics: Study counting techniques, permutations, and combinations.
• Graph Theory: Understand the basic concepts of graph theory and their applications in computer science and other fields.
• Number Theory: Learn about the properties of integers, divisibility, prime numbers, and modular arithmetic.

Strategies for Solving Different Types of Problems

Different types of mathematical problems require different problem-solving strategies. Here are some specific strategies for various types of problems:

1. Word Problems

Word problems present mathematical concepts in a real-world context. To solve word problems effectively:

• Read the Problem Carefully: Understand the context and identify the key information.
• Define Variables: Assign variables to represent unknown quantities.
• Write Equations: Translate the information in the problem into mathematical equations.
• Solve the Equations: Use algebraic techniques to solve the equations.
• Check Your Answer: Verify that the solution makes sense in the context of the problem.

2. Proof Problems

Proof problems require you to prove a mathematical statement using logical reasoning and mathematical principles. To solve proof problems:

• Understand the Statement: Clearly understand the statement you are trying to prove.
• Identify Assumptions: Determine the given information and assumptions.
• Choose a Proof Method: Select an appropriate proof method (direct proof, indirect proof, proof by contradiction, etc.).
• Write the Proof: Present your argument in a clear, logical, and step-by-step manner.
• Verify the Proof: Ensure that each step in your proof is justified and logically sound.

3. Optimization Problems

Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints. To solve optimization problems:

• Define the Objective Function: Identify the function you want to maximize or minimize.
• Identify Constraints: Determine the constraints that limit the possible values of the variables.
• Set Up the Problem: Formulate the problem mathematically, using equations and inequalities.
• Solve the Problem: Use calculus or other optimization techniques to find the optimal solution.
• Verify the Solution: Check that the solution satisfies the constraints and provides the desired maximum or minimum value.

4. Geometry Problems

Geometry problems often involve diagrams and require you to apply geometric principles and formulas. To solve geometry problems:

• Draw a Diagram: If a diagram is not provided, draw one yourself.
• Label the Diagram: Label all known and unknown quantities on the diagram.
• Identify Geometric Relationships: Recognize geometric relationships such as congruence, similarity, and parallelism.
• Apply Geometric Theorems: Use relevant geometric theorems and formulas to solve the problem.
• Check Your Answer: Verify that the solution is reasonable and consistent with the given information.

Common Mistakes and How to Avoid Them

Even with a solid understanding of problem-solving strategies and mathematical principles, it's easy to make mistakes. Here are some common mistakes and tips for avoiding them:

• Misunderstanding the Problem: Read the problem carefully and make sure you understand what is being asked.
• Making Calculation Errors: Double-check your calculations and use a calculator if necessary.
• Applying the Wrong Formula: Ensure that you are using the correct formula for the given problem.
• Ignoring Constraints: Pay attention to any constraints or conditions given in the problem.
• Not Checking Your Answer: Always check your answer to make sure it is reasonable and accurate.

Practice and Resources

Practice is essential for improving your problem-solving skills. Here are some resources that can help you practice:

• Textbooks: Work through the examples and exercises in your math textbooks.
• Online Resources: Use online resources such as Khan Academy, Coursera, and edX to learn new concepts and practice problems.
• Practice Problems: Solve a variety of practice problems from different sources.
• Study Groups: Join a study group and work with others to solve problems and discuss concepts.

Conclusion

Solving mathematical problems is a skill that can be developed through practice and a structured approach. By understanding the problem-solving process, mastering mathematical principles and techniques, and using effective strategies, you can become a confident and successful problem solver. Remember to approach problems systematically, check your work, and learn from your mistakes. With dedication and effort, you can master the art of mathematical problem-solving and unlock the power of mathematics in various aspects of your life.

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After applying the distributive property (FOIL method) to the expression $(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7})$, we obtain the result $\sqrt{10} + \sqrt{15} - \sqrt{14} - \sqrt{21}$. This matches option B, which is therefore the correct solution. This article has thoroughly explored the process of simplifying expressions with square roots, providing a comprehensive understanding of the steps and techniques involved. By applying these principles, you can confidently tackle similar mathematical challenges.