What Is The Following Quotient? 2 13 + 11 \frac{2}{\sqrt{13}+\sqrt{11}} 13 ​ + 11 ​ 2 ​ A. 13 − 11 \sqrt{13}-\sqrt{11} 13 ​ − 11 ​ B. 13 + 11 6 \frac{\sqrt{13}+\sqrt{11}}{6} 6 13 ​ + 11 ​ ​ C. 13 + 11 12 \frac{\sqrt{13}+\sqrt{11}}{12} 12 13 ​ + 11 ​ ​ D. 13 − 2 11 \sqrt{13}-2 \sqrt{11} 13 ​ − 2 11 ​

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Rationalizing the Denominator

When dealing with expressions that involve square roots in the denominator, it's essential to rationalize the denominator to simplify the expression. Rationalizing the denominator involves multiplying both the numerator and the denominator by a specific value to eliminate the square root from the denominator.

In this case, we're given the expression 213+11\frac{2}{\sqrt{13}+\sqrt{11}}. Our goal is to simplify this expression by rationalizing the denominator.

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. The conjugate of a binomial expression a+ba + b is a−ba - b. In this case, the conjugate of 13+11\sqrt{13} + \sqrt{11} is 13−11\sqrt{13} - \sqrt{11}.

$\frac{2}{\sqrt{13}+\sqrt{11}} \cdot \frac{\sqrt{13}-\sqrt{11}}{\sqrt{13}-\sqrt{11}}$

Step 2: Simplify the Expression

Now that we've multiplied the numerator and denominator by the conjugate, we can simplify the expression.

$\frac{2(\sqrt{13}-\sqrt{11})}{(\sqrt{13}+\sqrt{11})(\sqrt{13}-\sqrt{11})}$

Step 3: Apply the Difference of Squares Formula

The denominator can be simplified using the difference of squares formula: (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2.

$\frac{2(\sqrt{13}-\sqrt{11})}{(\sqrt{13})^2 - (\sqrt{11})^2}$

Step 4: Simplify the Expression Further

Now that we've applied the difference of squares formula, we can simplify the expression further.

$\frac{2(\sqrt{13}-\sqrt{11})}{13 - 11}$

Step 5: Simplify the Fraction

The fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2.

$\frac{2(\sqrt{13}-\sqrt{11})}{2}$

Step 6: Final Simplification

Now that we've simplified the fraction, we can simplify the expression further.

$\sqrt{13}-\sqrt{11}$

The final answer is 13−11\boxed{\sqrt{13}-\sqrt{11}}.

Conclusion

In this article, we've shown how to rationalize the denominator of a given expression and simplify it to its final form. By following the steps outlined above, we can simplify expressions that involve square roots in the denominator. The key concept is to multiply both the numerator and the denominator by the conjugate of the denominator to eliminate the square root from the denominator.

Frequently Asked Questions

  • What is rationalizing the denominator? Rationalizing the denominator involves multiplying both the numerator and the denominator by a specific value to eliminate the square root from the denominator.
  • Why do we need to rationalize the denominator? We need to rationalize the denominator to simplify the expression and eliminate the square root from the denominator.
  • What is the conjugate of a binomial expression? The conjugate of a binomial expression a+ba + b is a−ba - b.

Final Thoughts

Rationalizing the denominator is an essential concept in algebra that allows us to simplify expressions that involve square roots in the denominator. By following the steps outlined above, we can simplify expressions and eliminate the square root from the denominator. This concept is crucial in solving problems that involve square roots and is a fundamental concept in algebra.

Frequently Asked Questions

Q: What is rationalizing the denominator?

A: Rationalizing the denominator involves multiplying both the numerator and the denominator by a specific value to eliminate the square root from the denominator.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to simplify the expression and eliminate the square root from the denominator. This is especially important when dealing with expressions that involve square roots in the denominator, as it allows us to simplify the expression and make it easier to work with.

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 3+43 + 4 is 3−43 - 4.

Q: How do I rationalize the denominator of an expression?

A: To rationalize the denominator of an expression, you need to multiply both the numerator and the denominator by the conjugate of the denominator. This will eliminate the square root from the denominator and simplify the expression.

Q: What is the difference of squares formula?

A: The difference of squares formula is (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. This formula can be used to simplify expressions that involve the product of two binomials.

Q: How do I apply the difference of squares formula?

A: To apply the difference of squares formula, you need to multiply the two binomials together and then simplify the resulting expression. For example, (3+4)(3−4)=32−42=9−16=−7(3+4)(3-4) = 3^2 - 4^2 = 9 - 16 = -7.

Q: What is the final answer to the original problem?

A: The final answer to the original problem is 13−11\boxed{\sqrt{13}-\sqrt{11}}.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is important because it allows us to simplify expressions that involve square roots in the denominator. This is especially important in algebra, where expressions with square roots in the denominator can be difficult to work with.

Q: Can I rationalize the denominator of any expression?

A: Yes, you can rationalize the denominator of any expression that involves a square root in the denominator. However, you need to make sure that you multiply both the numerator and the denominator by the conjugate of 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 errors in the calculation

Conclusion

Rationalizing the denominator is an essential concept in algebra that allows us to simplify expressions that involve square roots in the denominator. By following the steps outlined above and avoiding common mistakes, you can rationalize the denominator of any expression and simplify it to its final form.

Final Thoughts

Rationalizing the denominator is a powerful tool that can be used to simplify expressions and make them easier to work with. By mastering this concept, you can solve problems that involve square roots and simplify expressions that would otherwise be difficult to work with.

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