Which Choice Is Equivalent To The Fraction Below When $x$ Is An Appropriate Value? Hint: Rationalize The Denominator And Simplify. $\frac{\sqrt{12}}{\sqrt{3}-3}$A. $-1+\sqrt{3}$B. $-1-\sqrt{3}$C.

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Rationalizing the Denominator: A Step-by-Step Guide to Simplifying Complex Fractions

When dealing with complex fractions, rationalizing the denominator is a crucial step in simplifying the expression. In this article, we will explore how to rationalize the denominator of a fraction and simplify it to its simplest form. We will use the given fraction 123−3\frac{\sqrt{12}}{\sqrt{3}-3} as an example and show how to simplify it to one of the given choices.

The given fraction is 123−3\frac{\sqrt{12}}{\sqrt{3}-3}. To simplify this fraction, we need to rationalize the denominator, which means removing the radical from the denominator. To do this, we will multiply the numerator and denominator by the conjugate of the denominator.

The conjugate of the denominator 3−3\sqrt{3}-3 is 3+3\sqrt{3}+3. To rationalize the denominator, we will multiply the numerator and denominator by the conjugate:

123−3⋅3+33+3\frac{\sqrt{12}}{\sqrt{3}-3} \cdot \frac{\sqrt{3}+3}{\sqrt{3}+3}

Using the distributive property, we can expand the numerator and denominator:

12(3+3)(3−3)(3+3)\frac{\sqrt{12}(\sqrt{3}+3)}{(\sqrt{3}-3)(\sqrt{3}+3)}

Now, we can simplify the numerator and denominator separately:

Simplifying the Numerator

The numerator is 12(3+3)\sqrt{12}(\sqrt{3}+3). We can simplify this expression by multiplying the square root of 12 by the square root of 3 and 3:

12(3+3)=36+312\sqrt{12}(\sqrt{3}+3) = \sqrt{36} + 3\sqrt{12}

Since 36=6\sqrt{36} = 6, we can simplify the numerator further:

36+312=6+312\sqrt{36} + 3\sqrt{12} = 6 + 3\sqrt{12}

Simplifying the Denominator

The denominator is (3−3)(3+3)(\sqrt{3}-3)(\sqrt{3}+3). We can simplify this expression by multiplying the two binomials:

(3−3)(3+3)=3−9(\sqrt{3}-3)(\sqrt{3}+3) = 3 - 9

Since 3−9=−63 - 9 = -6, we can simplify the denominator further:

(3−3)(3+3)=−6(\sqrt{3}-3)(\sqrt{3}+3) = -6

Simplifying the Fraction

Now that we have simplified the numerator and denominator, we can simplify the fraction:

12(3+3)(3−3)(3+3)=6+312−6\frac{\sqrt{12}(\sqrt{3}+3)}{(\sqrt{3}-3)(\sqrt{3}+3)} = \frac{6 + 3\sqrt{12}}{-6}

To simplify this fraction, we can divide the numerator by the denominator:

6+312−6=−1−122\frac{6 + 3\sqrt{12}}{-6} = -1 - \frac{\sqrt{12}}{2}

Simplifying the Square Root

The square root of 12 can be simplified by finding the largest perfect square that divides 12:

12=4⋅3=23\sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3}

Substituting this expression into the fraction, we get:

−1−122=−1−232-1 - \frac{\sqrt{12}}{2} = -1 - \frac{2\sqrt{3}}{2}

Simplifying the fraction, we get:

−1−232=−1−3-1 - \frac{2\sqrt{3}}{2} = -1 - \sqrt{3}

In this article, we have shown how to rationalize the denominator of a fraction and simplify it to its simplest form. We used the given fraction 123−3\frac{\sqrt{12}}{\sqrt{3}-3} as an example and simplified it to −1−3-1 - \sqrt{3}. This result matches one of the given choices, which is B. −1−3-1-\sqrt{3}.

The final answer is B. −1−3-1-\sqrt{3}.
Rationalizing the Denominator: A Q&A Guide

In our previous article, we explored how to rationalize the denominator of a fraction and simplify it to its simplest form. In this article, we will answer some common questions related to rationalizing the denominator and provide additional examples to help you understand the concept.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is the process of removing the radical from the denominator of a fraction. This is done by multiplying the numerator and denominator by the conjugate of the denominator.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to simplify the fraction and make it easier to work with. Rationalizing the denominator also helps to eliminate any radicals in the denominator, which can make the fraction more manageable.

Q: How do I find the conjugate of the denominator?

A: The conjugate of the denominator is found by changing the sign of the second term in the denominator. For example, if the denominator is 3−3\sqrt{3}-3, the conjugate is 3+3\sqrt{3}+3.

Q: What is the difference between rationalizing the denominator and simplifying the fraction?

A: Rationalizing the denominator and simplifying the fraction are two separate processes. Rationalizing the denominator involves removing the radical from the denominator, while simplifying the fraction involves reducing the fraction to its simplest form.

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. However, you will need to follow the same steps as before, but with the negative exponent in mind.

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

A: To rationalize the denominator of a fraction with a binomial denominator, you will need to multiply the numerator and denominator by the conjugate of the denominator. This will eliminate the radical from the denominator.

Q: Can I use rationalizing the denominator to simplify a fraction with a radical in the numerator?

A: Yes, you can use rationalizing the denominator to simplify a fraction with a radical in the numerator. However, you will need to follow the same steps as before, but with the radical in the numerator in mind.

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 the numerator and denominator by the conjugate of the denominator
  • Not simplifying the fraction after rationalizing the denominator
  • Not checking for any remaining radicals in the denominator

In this article, we have answered some common questions related to rationalizing the denominator and provided additional examples to help you understand the concept. We hope that this article has been helpful in clarifying any confusion you may have had about rationalizing the denominator.

  • Always multiply the numerator and denominator by the conjugate of the denominator to rationalize the denominator.
  • Simplify the fraction after rationalizing the denominator to ensure that it is in its simplest form.
  • Check for any remaining radicals in the denominator to ensure that the fraction is fully simplified.
  • For more information on rationalizing the denominator, check out our previous article on the topic.
  • For additional examples and practice problems, check out our online resources and practice exercises.

The final answer is B. −1−3-1-\sqrt{3}.