Answer :
To identify the correct equivalent expression for the fraction [tex]\(\frac{\sqrt{12}}{\sqrt{3} - 3}\)[/tex] by rationalizing the denominator and simplifying, follow these steps:
1. Simplify the numerator:
[tex]\[\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}.\][/tex]
So, the fraction becomes:
[tex]\[ \frac{2\sqrt{3}}{\sqrt{3} - 3} \][/tex]
2. Rationalize the denominator:
To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, [tex]\(\sqrt{3} + 3\)[/tex]:
[tex]\[ \frac{2\sqrt{3}}{\sqrt{3} - 3} \times \frac{\sqrt{3} + 3}{\sqrt{3} + 3} \][/tex]
This multiplication will eliminate the square root in the denominator.
3. Perform the multiplication in the numerator:
[tex]\[ 2\sqrt{3} \times (\sqrt{3} + 3) = 2\sqrt{3} \times \sqrt{3} + 2\sqrt{3} \times 3 \][/tex]
[tex]\[ = 2 \times 3 + 6\sqrt{3} \][/tex]
[tex]\[ = 6 + 6\sqrt{3} \][/tex]
4. Perform the multiplication in the denominator:
[tex]\[ (\sqrt{3} - 3) \times (\sqrt{3} + 3) = (\sqrt{3})^2 - (3)^2 \][/tex]
[tex]\[ = 3 - 9 \][/tex]
[tex]\[ = -6 \][/tex]
5. Combine the results:
[tex]\[ \frac{6 + 6\sqrt{3}}{-6} \][/tex]
Treat the numerator as the sum of two terms:
[tex]\[ \frac{6}{-6} + \frac{6\sqrt{3}}{-6} \][/tex]
Simplify each term:
[tex]\[ \frac{6}{-6} = -1 \][/tex]
and
[tex]\[ \frac{6\sqrt{3}}{-6} = -\sqrt{3} \][/tex]
6. Combine the simplified terms:
[tex]\[ -1 - \sqrt{3} \][/tex]
Thus, the simplified expression for [tex]\(\frac{\sqrt{12}}{\sqrt{3} - 3}\)[/tex] is [tex]\(-1 - \sqrt{3}\)[/tex], which corresponds to choice C.
Therefore, the correct answer is [tex]\(\boxed{C}\)[/tex].
1. Simplify the numerator:
[tex]\[\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}.\][/tex]
So, the fraction becomes:
[tex]\[ \frac{2\sqrt{3}}{\sqrt{3} - 3} \][/tex]
2. Rationalize the denominator:
To rationalize the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, [tex]\(\sqrt{3} + 3\)[/tex]:
[tex]\[ \frac{2\sqrt{3}}{\sqrt{3} - 3} \times \frac{\sqrt{3} + 3}{\sqrt{3} + 3} \][/tex]
This multiplication will eliminate the square root in the denominator.
3. Perform the multiplication in the numerator:
[tex]\[ 2\sqrt{3} \times (\sqrt{3} + 3) = 2\sqrt{3} \times \sqrt{3} + 2\sqrt{3} \times 3 \][/tex]
[tex]\[ = 2 \times 3 + 6\sqrt{3} \][/tex]
[tex]\[ = 6 + 6\sqrt{3} \][/tex]
4. Perform the multiplication in the denominator:
[tex]\[ (\sqrt{3} - 3) \times (\sqrt{3} + 3) = (\sqrt{3})^2 - (3)^2 \][/tex]
[tex]\[ = 3 - 9 \][/tex]
[tex]\[ = -6 \][/tex]
5. Combine the results:
[tex]\[ \frac{6 + 6\sqrt{3}}{-6} \][/tex]
Treat the numerator as the sum of two terms:
[tex]\[ \frac{6}{-6} + \frac{6\sqrt{3}}{-6} \][/tex]
Simplify each term:
[tex]\[ \frac{6}{-6} = -1 \][/tex]
and
[tex]\[ \frac{6\sqrt{3}}{-6} = -\sqrt{3} \][/tex]
6. Combine the simplified terms:
[tex]\[ -1 - \sqrt{3} \][/tex]
Thus, the simplified expression for [tex]\(\frac{\sqrt{12}}{\sqrt{3} - 3}\)[/tex] is [tex]\(-1 - \sqrt{3}\)[/tex], which corresponds to choice C.
Therefore, the correct answer is [tex]\(\boxed{C}\)[/tex].
