Simplify The Expression:${ \frac{2 \sqrt{2} + 3 \sqrt{2}}{5 \sqrt{5} - 4 \sqrt{6}} }$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. One common expression that requires simplification is the one given above: 22+3255−46\frac{2 \sqrt{2} + 3 \sqrt{2}}{5 \sqrt{5} - 4 \sqrt{6}}. In this article, we will break down the steps to simplify this expression and provide a clear understanding of the process.

Understanding the Expression

Before we start simplifying the expression, let's analyze it. We have a fraction with two square roots in the numerator and two square roots in the denominator. The numerator is 22+322 \sqrt{2} + 3 \sqrt{2}, and the denominator is 55−465 \sqrt{5} - 4 \sqrt{6}. Our goal is to simplify this expression by rationalizing the denominator.

Rationalizing the Denominator

Rationalizing the denominator means removing the square roots from the denominator. To do this, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial expression a+ba + b is a−ba - b. In this case, the conjugate of 55−465 \sqrt{5} - 4 \sqrt{6} is 55+465 \sqrt{5} + 4 \sqrt{6}.

Step 1: Multiply the Numerator and Denominator by the Conjugate

To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. This will eliminate the square roots from the denominator.

22+3255−46⋅55+4655+46\frac{2 \sqrt{2} + 3 \sqrt{2}}{5 \sqrt{5} - 4 \sqrt{6}} \cdot \frac{5 \sqrt{5} + 4 \sqrt{6}}{5 \sqrt{5} + 4 \sqrt{6}}

Step 4: Simplify the Expression

Now that we have multiplied the numerator and denominator by the conjugate, we can simplify the expression. We will start by multiplying the numerators and denominators separately.

Numerator: (22+32)⋅(55+46)(2 \sqrt{2} + 3 \sqrt{2}) \cdot (5 \sqrt{5} + 4 \sqrt{6}) Denominator: (55−46)⋅(55+46)(5 \sqrt{5} - 4 \sqrt{6}) \cdot (5 \sqrt{5} + 4 \sqrt{6})

Step 5: Apply the Distributive Property

To simplify the expression, we need to apply the distributive property to both the numerator and the denominator.

Numerator: 22⋅55+22⋅46+32⋅55+32⋅462 \sqrt{2} \cdot 5 \sqrt{5} + 2 \sqrt{2} \cdot 4 \sqrt{6} + 3 \sqrt{2} \cdot 5 \sqrt{5} + 3 \sqrt{2} \cdot 4 \sqrt{6} Denominator: 55⋅55−46⋅55+55⋅46−46⋅465 \sqrt{5} \cdot 5 \sqrt{5} - 4 \sqrt{6} \cdot 5 \sqrt{5} + 5 \sqrt{5} \cdot 4 \sqrt{6} - 4 \sqrt{6} \cdot 4 \sqrt{6}

Step 6: Simplify the Numerator and Denominator

Now that we have applied the distributive property, we can simplify the numerator and denominator separately.

Numerator: 1010+812+1510+121210 \sqrt{10} + 8 \sqrt{12} + 15 \sqrt{10} + 12 \sqrt{12} Denominator: 2525−2030+2030−163625 \sqrt{25} - 20 \sqrt{30} + 20 \sqrt{30} - 16 \sqrt{36}

Step 7: Combine Like Terms

To simplify the expression further, we need to combine like terms in the numerator and denominator.

Numerator: (1010+1510)+(812+1212)(10 \sqrt{10} + 15 \sqrt{10}) + (8 \sqrt{12} + 12 \sqrt{12}) Denominator: 2525−163625 \sqrt{25} - 16 \sqrt{36}

Step 8: Simplify the Square Roots

Now that we have combined like terms, we can simplify the square roots in the numerator and denominator.

Numerator: 2510+201225 \sqrt{10} + 20 \sqrt{12} Denominator: 2525−163625 \sqrt{25} - 16 \sqrt{36}

Step 9: Simplify the Expression

Finally, we can simplify the expression by evaluating the square roots in the numerator and denominator.

Numerator: 2510+201225 \sqrt{10} + 20 \sqrt{12} Denominator: 25⋅5−16⋅625 \cdot 5 - 16 \cdot 6

Step 10: Evaluate the Expression

Now that we have simplified the expression, we can evaluate it.

Numerator: 2510+201225 \sqrt{10} + 20 \sqrt{12} Denominator: 125−96125 - 96

Step 11: Simplify the Expression

Finally, we can simplify the expression by evaluating the numerator and denominator.

Numerator: 2510+201225 \sqrt{10} + 20 \sqrt{12} Denominator: 2929

Conclusion

In this article, we have simplified the expression 22+3255−46\frac{2 \sqrt{2} + 3 \sqrt{2}}{5 \sqrt{5} - 4 \sqrt{6}} by rationalizing the denominator. We have broken down the steps to simplify the expression and provided a clear understanding of the process. By following these steps, we can simplify complex expressions and solve problems efficiently.

Final Answer

The final answer is 2510+201229\boxed{\frac{25 \sqrt{10} + 20 \sqrt{12}}{29}}.

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Introduction

In our previous article, we simplified the expression 22+3255−46\frac{2 \sqrt{2} + 3 \sqrt{2}}{5 \sqrt{5} - 4 \sqrt{6}} by rationalizing the denominator. In this article, we will provide a Q&A guide to help you understand the process of simplifying expressions with square roots.

Q&A Guide

Q: What is rationalizing the denominator?

A: Rationalizing the denominator means removing the square roots from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by the conjugate of the denominator.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression a+ba + b is a−ba - b. For example, the conjugate of 55−465 \sqrt{5} - 4 \sqrt{6} is 55+465 \sqrt{5} + 4 \sqrt{6}.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to simplify the expression and make it easier to work with. Rationalizing the denominator helps us to eliminate the square roots from the denominator, which makes it easier to evaluate the expression.

Q: How do we rationalize the denominator?

A: To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. This will eliminate the square roots from the denominator.

Q: What are some common mistakes to avoid when rationalizing the denominator?

A: Some common mistakes to avoid when rationalizing the denominator include:

  • Not multiplying both the numerator and the denominator by the conjugate of the denominator
  • Not simplifying the expression after rationalizing the denominator
  • Not checking for any remaining square roots in the denominator

Q: How do we simplify the expression after rationalizing the denominator?

A: To simplify the expression after rationalizing the denominator, we need to combine like terms in the numerator and denominator. We also need to simplify any remaining square roots in the numerator and denominator.

Q: What are some examples of expressions that require rationalizing the denominator?

A: Some examples of expressions that require rationalizing the denominator include:

  • 22+3255−46\frac{2 \sqrt{2} + 3 \sqrt{2}}{5 \sqrt{5} - 4 \sqrt{6}}
  • 43−2332+22\frac{4 \sqrt{3} - 2 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{2}}
  • 35+2523−33\frac{3 \sqrt{5} + 2 \sqrt{5}}{2 \sqrt{3} - 3 \sqrt{3}}

Q: How do we evaluate the expression after rationalizing the denominator?

A: To evaluate the expression after rationalizing the denominator, we need to simplify the numerator and denominator separately. We also need to check for any remaining square roots in the denominator.

Conclusion

In this article, we have provided a Q&A guide to help you understand the process of simplifying expressions with square roots. We have covered common mistakes to avoid, examples of expressions that require rationalizing the denominator, and how to evaluate the expression after rationalizing the denominator. By following these steps, you can simplify complex expressions and solve problems efficiently.

Final Answer

The final answer is 2510+201229\boxed{\frac{25 \sqrt{10} + 20 \sqrt{12}}{29}}.