Solved on Nov 29, 2023

Simplify the fraction $\frac{\sqrt{12}}{\sqrt{3}-3}$ and choose the equivalent expression.

STEP 1

Assumptions
1. We are given the fraction $\frac{\sqrt{12}}{\sqrt{3}-3}$
2. We need to rationalize the denominator and simplify the fraction
3. $x$ is an appropriate value

STEP 2

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{3}-3$ is $\sqrt{3}+3$ .
$\frac{\sqrt{12}}{\sqrt{3}-3} \times \frac{\sqrt{3}+3}{\sqrt{3}+3}$

STEP 3

Multiply the numerators and the denominators separately.
$\frac{\sqrt{12}(\sqrt{3}+3)}{(\sqrt{3}-3)(\sqrt{3}+3)}$

STEP 4

Use the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$ , to simplify the denominator.
$\frac{\sqrt{12}(\sqrt{3}+3)}{3-9}$

STEP 5

Calculate the denominator.
$\frac{\sqrt{12}(\sqrt{3}+3)}{-6}$

STEP 6

Multiply the numerator.
$\frac{\sqrt{36}+3\sqrt{12}}{-6}$

STEP 7

Simplify the square root in the numerator.
$\frac{6+3\sqrt{12}}{-6}$

STEP 8

Simplify the fraction by dividing each term in the numerator by -6.
$-1-\frac{\sqrt{12}}{2}$

STEP 9

Simplify the square root in the numerator.
$-1-\sqrt{3}$
So, the equivalent form of the given fraction is $-1-\sqrt{3}$ .
Therefore, the correct choice is D. $-1-\sqrt{3}$ .

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Simplify the fraction $\frac{\sqrt{12}}{\sqrt{3}-3}$ and choose the equivalent expression.

STEP 1

Assumptions
1. We are given the fraction $\frac{\sqrt{12}}{\sqrt{3}-3}$
2. We need to rationalize the denominator and simplify the fraction
3. $x$ is an appropriate value

STEP 2

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{3}-3$ is $\sqrt{3}+3$ .
$\frac{\sqrt{12}}{\sqrt{3}-3} \times \frac{\sqrt{3}+3}{\sqrt{3}+3}$

STEP 3

Multiply the numerators and the denominators separately.
$\frac{\sqrt{12}(\sqrt{3}+3)}{(\sqrt{3}-3)(\sqrt{3}+3)}$

STEP 4

Use the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$ , to simplify the denominator.
$\frac{\sqrt{12}(\sqrt{3}+3)}{3-9}$

STEP 5

Calculate the denominator.
$\frac{\sqrt{12}(\sqrt{3}+3)}{-6}$

STEP 6

Multiply the numerator.
$\frac{\sqrt{36}+3\sqrt{12}}{-6}$

STEP 7

Simplify the square root in the numerator.
$\frac{6+3\sqrt{12}}{-6}$

STEP 8

Simplify the fraction by dividing each term in the numerator by -6.
$-1-\frac{\sqrt{12}}{2}$

STEP 9

Simplify the square root in the numerator.
$-1-\sqrt{3}$
So, the equivalent form of the given fraction is $-1-\sqrt{3}$ .
Therefore, the correct choice is D. $-1-\sqrt{3}$ .

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Simplify the fraction 123−3\frac{\sqrt{12}}{\sqrt{3}-3}3​−312​​ and choose the equivalent expression.

STEP 1

Assumptions1. We are given the fraction 123−3\frac{\sqrt{12}}{\sqrt{3}-3}3​−312​​2. We need to rationalize the denominator and simplify the fraction3. xxx is an appropriate value

STEP 2

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of 3−3\sqrt{3}-33​−3 is 3+3\sqrt{3}+33​+3.123−3×3+33+3 \frac{\sqrt{12}}{\sqrt{3}-3} \times \frac{\sqrt{3}+3}{\sqrt{3}+3} 3​−312​​×3​+33​+3​

STEP 3

Multiply the numerators and the denominators separately.12(3+3)(3−3)(3+3) \frac{\sqrt{12}(\sqrt{3}+3)}{(\sqrt{3}-3)(\sqrt{3}+3)} (3​−3)(3​+3)12​(3​+3)​

STEP 4

Use the difference of squares formula, a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b)a2−b2=(a−b)(a+b), to simplify the denominator.12(3+3)3−9 \frac{\sqrt{12}(\sqrt{3}+3)}{3-9} 3−912​(3​+3)​

STEP 5

Calculate the denominator.12(3+3)−6 \frac{\sqrt{12}(\sqrt{3}+3)}{-6} −612​(3​+3)​

STEP 6

Multiply the numerator.36+312−6 \frac{\sqrt{36}+3\sqrt{12}}{-6} −636​+312​​

STEP 7

Simplify the square root in the numerator.6+312−6 \frac{6+3\sqrt{12}}{-6} −66+312​​

STEP 8

Simplify the fraction by dividing each term in the numerator by -6.−1−122 -1-\frac{\sqrt{12}}{2} −1−212​​

STEP 9

Simplify the square root in the numerator.−1−3 -1-\sqrt{3} −1−3​So, the equivalent form of the given fraction is −1−3-1-\sqrt{3}−1−3​.Therefore, the correct choice is D. −1−3-1-\sqrt{3}−1−3​.

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Simplify the fraction $\frac{\sqrt{12}}{\sqrt{3}-3}$ and choose the equivalent expression.

Assumptions
1. We are given the fraction $\frac{\sqrt{12}}{\sqrt{3}-3}$
2. We need to rationalize the denominator and simplify the fraction
3. $x$ is an appropriate value

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{3}-3$ is $\sqrt{3}+3$ .
$\frac{\sqrt{12}}{\sqrt{3}-3} \times \frac{\sqrt{3}+3}{\sqrt{3}+3}$

Multiply the numerators and the denominators separately.
$\frac{\sqrt{12}(\sqrt{3}+3)}{(\sqrt{3}-3)(\sqrt{3}+3)}$

Use the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$ , to simplify the denominator.
$\frac{\sqrt{12}(\sqrt{3}+3)}{3-9}$

Calculate the denominator.
$\frac{\sqrt{12}(\sqrt{3}+3)}{-6}$

Multiply the numerator.
$\frac{\sqrt{36}+3\sqrt{12}}{-6}$

Simplify the square root in the numerator.
$\frac{6+3\sqrt{12}}{-6}$

Simplify the fraction by dividing each term in the numerator by -6.
$-1-\frac{\sqrt{12}}{2}$

Simplify the square root in the numerator.
$-1-\sqrt{3}$
So, the equivalent form of the given fraction is $-1-\sqrt{3}$ .
Therefore, the correct choice is D. $-1-\sqrt{3}$ .