Simplify $\frac{5}{\sqrt{3}+\sqrt{2}}+\frac{3}{\sqrt{3}-\sqrt{2}}$.

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Introduction

Simplifying complex fractions involving square roots can be a challenging task, especially when dealing with multiple terms and different denominators. In this article, we will explore a step-by-step approach to simplify the given expression: 53+2+33βˆ’2\frac{5}{\sqrt{3}+\sqrt{2}}+\frac{3}{\sqrt{3}-\sqrt{2}}. By applying various mathematical techniques, we will transform the given expression into a simpler form.

Understanding the Problem

The given expression consists of two fractions with different denominators, each containing square roots. To simplify this expression, we need to find a common denominator and then combine the two fractions. However, before we proceed with the simplification process, let's analyze the given expression and understand its structure.

The first fraction is 53+2\frac{5}{\sqrt{3}+\sqrt{2}}, where the denominator is the sum of two square roots. The second fraction is 33βˆ’2\frac{3}{\sqrt{3}-\sqrt{2}}, where the denominator is the difference of two square roots. Our goal is to simplify this expression by combining the two fractions.

Rationalizing the Denominators

To simplify the given expression, we need to rationalize the denominators of both fractions. Rationalizing the denominator involves multiplying the numerator and denominator by a conjugate of the denominator, which eliminates the square root term.

For the first fraction, the conjugate of the denominator 3+2\sqrt{3}+\sqrt{2} is 3βˆ’2\sqrt{3}-\sqrt{2}. We multiply the numerator and denominator of the first fraction by this conjugate:

53+2β‹…3βˆ’23βˆ’2\frac{5}{\sqrt{3}+\sqrt{2}} \cdot \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}}

This simplifies to:

5(3βˆ’2)(3+2)(3βˆ’2)\frac{5(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}

Using the difference of squares formula (a+b)(aβˆ’b)=a2βˆ’b2(a+b)(a-b) = a^2 - b^2, we can simplify the denominator:

(3+2)(3βˆ’2)=(3)2βˆ’(2)2=3βˆ’2=1(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1

Therefore, the first fraction simplifies to:

5(3βˆ’2)1=5(3βˆ’2)\frac{5(\sqrt{3}-\sqrt{2})}{1} = 5(\sqrt{3}-\sqrt{2})

Similarly, for the second fraction, the conjugate of the denominator 3βˆ’2\sqrt{3}-\sqrt{2} is 3+2\sqrt{3}+\sqrt{2}. We multiply the numerator and denominator of the second fraction by this conjugate:

33βˆ’2β‹…3+23+2\frac{3}{\sqrt{3}-\sqrt{2}} \cdot \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}}

This simplifies to:

3(3+2)(3βˆ’2)(3+2)\frac{3(\sqrt{3}+\sqrt{2})}{(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2})}

Using the difference of squares formula (a+b)(aβˆ’b)=a2βˆ’b2(a+b)(a-b) = a^2 - b^2, we can simplify the denominator:

(3βˆ’2)(3+2)=(3)2βˆ’(2)2=3βˆ’2=1(\sqrt{3}-\sqrt{2})(\sqrt{3}+\sqrt{2}) = (\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1

Therefore, the second fraction simplifies to:

3(3+2)1=3(3+2)\frac{3(\sqrt{3}+\sqrt{2})}{1} = 3(\sqrt{3}+\sqrt{2})

Combining the Fractions

Now that we have simplified both fractions, we can combine them by adding the two expressions:

5(3βˆ’2)+3(3+2)5(\sqrt{3}-\sqrt{2}) + 3(\sqrt{3}+\sqrt{2})

To add these two expressions, we need to combine like terms. We can do this by distributing the coefficients to the terms inside the parentheses:

53βˆ’52+33+325\sqrt{3} - 5\sqrt{2} + 3\sqrt{3} + 3\sqrt{2}

Now, we can combine the like terms:

(53+33)+(βˆ’52+32)(5\sqrt{3} + 3\sqrt{3}) + (-5\sqrt{2} + 3\sqrt{2})

This simplifies to:

83βˆ’228\sqrt{3} - 2\sqrt{2}

Therefore, the simplified expression is:

83βˆ’22\boxed{8\sqrt{3} - 2\sqrt{2}}

Conclusion

In this article, we simplified the given expression 53+2+33βˆ’2\frac{5}{\sqrt{3}+\sqrt{2}}+\frac{3}{\sqrt{3}-\sqrt{2}} by rationalizing the denominators and combining the fractions. By applying various mathematical techniques, we transformed the given expression into a simpler form. The final simplified expression is 83βˆ’228\sqrt{3} - 2\sqrt{2}.

Final Answer

The final answer is 83βˆ’22\boxed{8\sqrt{3} - 2\sqrt{2}}.

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Introduction

In our previous article, we simplified the given expression 53+2+33βˆ’2\frac{5}{\sqrt{3}+\sqrt{2}}+\frac{3}{\sqrt{3}-\sqrt{2}} by rationalizing the denominators and combining the fractions. In this article, we will answer some frequently asked questions related to the simplification process.

Q&A

Q: What is the purpose of rationalizing the denominators?

A: Rationalizing the denominators is a process of eliminating the square root term from the denominator. This is done by multiplying the numerator and denominator by a conjugate of the denominator, which is a term that, when multiplied by the denominator, eliminates the square root term.

Q: Why do we need to rationalize the denominators?

A: We need to rationalize the denominators because the square root terms in the denominators make it difficult to add or subtract the fractions. By rationalizing the denominators, we can eliminate the square root terms and make it easier to combine the fractions.

Q: How do we find the conjugate of a denominator?

A: To find the conjugate of a denominator, we need to change the sign of the middle term. For example, if the denominator is 3+2\sqrt{3}+\sqrt{2}, the conjugate is 3βˆ’2\sqrt{3}-\sqrt{2}.

Q: What is the difference of squares formula?

A: The difference of squares formula is (a+b)(aβˆ’b)=a2βˆ’b2(a+b)(a-b) = a^2 - b^2. This formula is used to simplify the denominator when rationalizing the denominators.

Q: How do we combine like terms?

A: To combine like terms, we need to distribute the coefficients to the terms inside the parentheses and then combine the like terms.

Q: What is the final simplified expression?

A: The final simplified expression is 83βˆ’228\sqrt{3} - 2\sqrt{2}.

Common Mistakes

Mistake 1: Not rationalizing the denominators

A: Not rationalizing the denominators can make it difficult to add or subtract the fractions. This can lead to incorrect answers.

Mistake 2: Not finding the conjugate of the denominator

A: Not finding the conjugate of the denominator can make it difficult to rationalize the denominator. This can lead to incorrect answers.

Mistake 3: Not combining like terms

A: Not combining like terms can lead to incorrect answers. This is because like terms are terms that have the same variable and coefficient.

Tips and Tricks

Tip 1: Always rationalize the denominators

A: Rationalizing the denominators is an important step in simplifying fractions. It makes it easier to add or subtract the fractions.

Tip 2: Find the conjugate of the denominator

A: Finding the conjugate of the denominator is an important step in rationalizing the denominators. It makes it easier to eliminate the square root term from the denominator.

Tip 3: Combine like terms

A: Combining like terms is an important step in simplifying fractions. It makes it easier to get the final answer.

Conclusion

In this article, we answered some frequently asked questions related to the simplification process of the given expression 53+2+33βˆ’2\frac{5}{\sqrt{3}+\sqrt{2}}+\frac{3}{\sqrt{3}-\sqrt{2}}. We also discussed some common mistakes and provided some tips and tricks to help you simplify fractions.

Final Answer

The final answer is 83βˆ’22\boxed{8\sqrt{3} - 2\sqrt{2}}.