Solve frac{3-2sqrt{12}}{1-3sqrt{3}}

$$\frac { 3 - 2 \sqrt { 12 } } { 1 - 3 \sqrt { 3 } }$$

Evaluate

$$\frac{33-5\sqrt{3}}{26}\approx 0.936144075$$

Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$\frac{3-2\times 2\sqrt{3}}{1-3\sqrt{3}}$$

Multiply -2 and 2 to get -4.

$$\frac{3-4\sqrt{3}}{1-3\sqrt{3}}$$

Rationalize the denominator of \frac{3-4\sqrt{3}}{1-3\sqrt{3}} by multiplying numerator and denominator by 1+3\sqrt{3}.

$$\frac{\left(3-4\sqrt{3}\right)\left(1+3\sqrt{3}\right)}{\left(1-3\sqrt{3}\right)\left(1+3\sqrt{3}\right)}$$

Consider \left(1-3\sqrt{3}\right)\left(1+3\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(3-4\sqrt{3}\right)\left(1+3\sqrt{3}\right)}{1^{2}-\left(-3\sqrt{3}\right)^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\left(3-4\sqrt{3}\right)\left(1+3\sqrt{3}\right)}{1-\left(-3\sqrt{3}\right)^{2}}$$

Expand \left(-3\sqrt{3}\right)^{2}.

$$\frac{\left(3-4\sqrt{3}\right)\left(1+3\sqrt{3}\right)}{1-\left(-3\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{\left(3-4\sqrt{3}\right)\left(1+3\sqrt{3}\right)}{1-9\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{\left(3-4\sqrt{3}\right)\left(1+3\sqrt{3}\right)}{1-9\times 3}$$

Multiply 9 and 3 to get 27.

$$\frac{\left(3-4\sqrt{3}\right)\left(1+3\sqrt{3}\right)}{1-27}$$

Subtract 27 from 1 to get -26.

$$\frac{\left(3-4\sqrt{3}\right)\left(1+3\sqrt{3}\right)}{-26}$$

Apply the distributive property by multiplying each term of 3-4\sqrt{3} by each term of 1+3\sqrt{3}.

$$\frac{3+9\sqrt{3}-4\sqrt{3}-12\left(\sqrt{3}\right)^{2}}{-26}$$

Combine 9\sqrt{3} and -4\sqrt{3} to get 5\sqrt{3}.

$$\frac{3+5\sqrt{3}-12\left(\sqrt{3}\right)^{2}}{-26}$$

The square of \sqrt{3} is 3.

$$\frac{3+5\sqrt{3}-12\times 3}{-26}$$

Multiply -12 and 3 to get -36.

$$\frac{3+5\sqrt{3}-36}{-26}$$

Subtract 36 from 3 to get -33.

$$\frac{-33+5\sqrt{3}}{-26}$$

Multiply both numerator and denominator by -1.

$$\frac{33-5\sqrt{3}}{26}$$