What Is The Following Quotient?A. 2 13 + 11 \frac{2}{\sqrt{13}+\sqrt{11}} 13 ​ + 11 ​ 2 ​ B. 13 − 2 11 \sqrt{13}-2 \sqrt{11} 13 ​ − 2 11 ​ C. 13 + 11 6 \frac{\sqrt{13}+\sqrt{11}}{6} 6 13 ​ + 11 ​ ​ D. 13 + 11 12 \frac{\sqrt{13}+\sqrt{11}}{12} 12 13 ​ + 11 ​ ​ E. 13 − 11 \sqrt{13}-\sqrt{11} 13 ​ − 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. This process 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}}, and we need to rationalize the denominator.

Rationalizing the Denominator: A Step-by-Step Guide

To rationalize the denominator, we'll 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}.

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

By multiplying the numerator and the denominator by the conjugate, we get:

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

Now, we can simplify the expression by multiplying the denominator:

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

Using the difference of squares formula, we can simplify the denominator further:

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

Simplifying the expression, we get:

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

Finally, we can simplify the expression by canceling out the common factor of 2:

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

Comparing the Result with the Options

Now that we've rationalized the denominator and simplified the expression, we can compare the result with the options provided:

  • A. 213+11\frac{2}{\sqrt{13}+\sqrt{11}}
  • B. 13−211\sqrt{13}-2 \sqrt{11}
  • C. 13+116\frac{\sqrt{13}+\sqrt{11}}{6}
  • D. 13+1112\frac{\sqrt{13}+\sqrt{11}}{12}
  • E. 13−11\sqrt{13}-\sqrt{11}

Based on our calculations, we can see that the correct answer is:

The Correct Answer is E. 13−11\sqrt{13}-\sqrt{11}

Conclusion

Rationalizing the denominator is an essential step in simplifying expressions that involve square roots in the denominator. By following the steps outlined in this article, we can simplify the expression 213+11\frac{2}{\sqrt{13}+\sqrt{11}} and arrive at the correct answer. Remember to always rationalize the denominator when dealing with expressions that involve square roots in the denominator.

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 is rationalizing the denominator important?

A: Rationalizing the denominator is essential in simplifying expressions that involve square roots in the denominator. It helps to eliminate the square root from the denominator, making it easier to work with the expression.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, 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.

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Q&A: Rationalizing the Denominator

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. This process is essential in simplifying expressions that involve square roots in the denominator.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is important because it helps to eliminate the square root from the denominator, making it easier to work with the expression. This is particularly useful when dealing with expressions that involve square roots in the denominator, as it allows us to simplify the expression and make it easier to evaluate.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, 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. For example, if we have the expression 213+11\frac{2}{\sqrt{13}+\sqrt{11}}, we would multiply both the numerator and the denominator by 13−11\sqrt{13}-\sqrt{11} to rationalize 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 13+11\sqrt{13} + \sqrt{11} is 13−11\sqrt{13} - \sqrt{11}.

Q: How do I simplify an expression with a rationalized denominator?

A: Once you have rationalized the denominator, you can simplify the expression by canceling out any common factors between the numerator and the denominator. For example, if we have the expression 2(13−11)2\frac{2(\sqrt{13}-\sqrt{11})}{2}, we can simplify it by canceling out the common factor of 2.

Q: Can I rationalize the denominator of a fraction with a negative exponent?

A: Yes, you can rationalize the denominator of a fraction with a negative exponent. To do this, you would multiply both the numerator and the denominator by the conjugate of the denominator, just like you would with a fraction with a positive exponent.

Q: How do I rationalize the denominator of a fraction with a variable in the denominator?

A: To rationalize the denominator of a fraction with a variable in the denominator, you would multiply both the numerator and the denominator by the conjugate of the denominator. For example, if we have the expression 2x+y\frac{2}{\sqrt{x}+\sqrt{y}}, we would multiply both the numerator and the denominator by x−y\sqrt{x}-\sqrt{y} to rationalize the denominator.

Q: Can I rationalize the denominator of a fraction with a complex number in the denominator?

A: Yes, you can rationalize the denominator of a fraction with a complex number in the denominator. To do this, you would multiply both the numerator and the denominator by the conjugate of the denominator. For example, if we have the expression 23+4i\frac{2}{3+4i}, we would multiply both the numerator and the denominator by 3−4i3-4i to rationalize the denominator.

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