Simplify The Expression:$(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7}$\]Options:A. $\sqrt{5}+\sqrt{-2}$B. $\sqrt{10}+\sqrt{15}-\sqrt{14}-\sqrt{21}$C. $2 \sqrt{5}-2 \sqrt{7}$D. $2 \sqrt{5}+3 \sqrt{5}-2 \sqrt{7}-3

by ADMIN 205 views

Introduction

In this article, we will delve into the world of algebra and focus on simplifying a given expression involving square roots. The expression we will be working with is (2+3)(5−7)(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7}). Our goal is to simplify this expression and arrive at a more manageable form. We will break down the solution into manageable steps, making it easier to understand and follow along.

Step 1: Understand the Expression

The given expression is a product of two binomials, each containing a square root term. To simplify this expression, we will use the distributive property, which states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac. We will apply this property to expand the given expression.

Step 2: Apply the Distributive Property

Using the distributive property, we can expand the given expression as follows:

(2+3)(5−7)=2(5−7)+3(5−7)(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7}) = \sqrt{2}(\sqrt{5}-\sqrt{7}) + \sqrt{3}(\sqrt{5}-\sqrt{7})

Step 3: Simplify Each Term

Now, we will simplify each term separately. We will start with the first term, 2(5−7)\sqrt{2}(\sqrt{5}-\sqrt{7}). Using the distributive property again, we can expand this term as follows:

2(5−7)=25−27\sqrt{2}(\sqrt{5}-\sqrt{7}) = \sqrt{2}\sqrt{5} - \sqrt{2}\sqrt{7}

Similarly, we will simplify the second term, 3(5−7)\sqrt{3}(\sqrt{5}-\sqrt{7}), as follows:

3(5−7)=35−37\sqrt{3}(\sqrt{5}-\sqrt{7}) = \sqrt{3}\sqrt{5} - \sqrt{3}\sqrt{7}

Step 4: Combine Like Terms

Now that we have simplified each term, we can combine like terms to arrive at the final expression. We will start by combining the like terms in the first term:

25−27=10−14\sqrt{2}\sqrt{5} - \sqrt{2}\sqrt{7} = \sqrt{10} - \sqrt{14}

Similarly, we will combine the like terms in the second term:

35−37=15−21\sqrt{3}\sqrt{5} - \sqrt{3}\sqrt{7} = \sqrt{15} - \sqrt{21}

Step 5: Combine the Terms

Finally, we will combine the two simplified terms to arrive at the final expression:

(2+3)(5−7)=(10−14)+(15−21)(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7}) = (\sqrt{10} - \sqrt{14}) + (\sqrt{15} - \sqrt{21})

Simplifying the Expression

To simplify the expression further, we can combine like terms:

(10−14)+(15−21)=10+15−14−21(\sqrt{10} - \sqrt{14}) + (\sqrt{15} - \sqrt{21}) = \sqrt{10} + \sqrt{15} - \sqrt{14} - \sqrt{21}

Conclusion

In this article, we simplified the given expression (2+3)(5−7)(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7}) using the distributive property and combining like terms. We arrived at the final expression 10+15−14−21\sqrt{10} + \sqrt{15} - \sqrt{14} - \sqrt{21}. This expression is the simplified form of the given expression.

Answer

The correct answer is:

  • B. 10+15−14−21\sqrt{10}+\sqrt{15}-\sqrt{14}-\sqrt{21}

Discussion

Introduction

In our previous article, we simplified the expression (2+3)(5−7)(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7}) using the distributive property and combining like terms. We arrived at the final expression 10+15−14−21\sqrt{10} + \sqrt{15} - \sqrt{14} - \sqrt{21}. In this article, we will provide a Q&A guide to help students understand the solution and to clarify any doubts they may have.

Q: What is the distributive property?

A: The distributive property is a fundamental concept in algebra that states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac. This property allows us to expand expressions involving multiple terms.

Q: How do I apply the distributive property to the given expression?

A: To apply the distributive property to the given expression, we need to multiply each term in the first binomial by each term in the second binomial. This will result in four separate terms, which we can then simplify.

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variable and exponent. Unlike terms are terms that have different variables or exponents. In the given expression, we can combine like terms to simplify the expression.

Q: How do I combine like terms?

A: To combine like terms, we need to add or subtract the coefficients of the like terms. For example, in the expression 10−14\sqrt{10} - \sqrt{14}, we can combine the like terms by adding the coefficients: 10+(−14)\sqrt{10} + (-\sqrt{14}).

Q: What is the final expression?

A: The final expression is 10+15−14−21\sqrt{10} + \sqrt{15} - \sqrt{14} - \sqrt{21}.

Q: Why is it important to simplify expressions?

A: Simplifying expressions is important because it helps us to:

  • Reduce the complexity of the expression
  • Make it easier to work with
  • Identify patterns and relationships between terms
  • Solve problems more efficiently

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include:

  • Forgetting to apply the distributive property
  • Not combining like terms
  • Making errors when adding or subtracting coefficients
  • Not checking the final expression for accuracy

Conclusion

In this article, we provided a Q&A guide to help students understand the solution to the expression (2+3)(5−7)(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7}). We clarified common doubts and provided tips for simplifying expressions. By following these tips and practicing regularly, students can develop their algebraic skills and become more confident in their ability to simplify expressions.

Frequently Asked Questions

  • Q: What is the distributive property? A: The distributive property is a fundamental concept in algebra that states that for any real numbers aa, bb, and cc, a(b+c)=ab+aca(b+c) = ab + ac.
  • Q: How do I apply the distributive property to the given expression? A: To apply the distributive property to the given expression, we need to multiply each term in the first binomial by each term in the second binomial.
  • Q: What is the difference between like terms and unlike terms? A: Like terms are terms that have the same variable and exponent. Unlike terms are terms that have different variables or exponents.
  • Q: How do I combine like terms? A: To combine like terms, we need to add or subtract the coefficients of the like terms.

Additional Resources

  • Algebraic Expressions: A comprehensive guide to algebraic expressions, including definitions, examples, and practice problems.
  • Distributive Property: A detailed explanation of the distributive property, including examples and practice problems.
  • Like Terms: A guide to like terms, including definitions, examples, and practice problems.

Conclusion

In this article, we provided a Q&A guide to help students understand the solution to the expression (2+3)(5−7)(\sqrt{2}+\sqrt{3})(\sqrt{5}-\sqrt{7}). We clarified common doubts and provided tips for simplifying expressions. By following these tips and practicing regularly, students can develop their algebraic skills and become more confident in their ability to simplify expressions.