Solve frac{3sqrt{3}}{6-3sqrt{6}}

$$\frac{ 3 \sqrt{ 3 } }{ 6-3 \sqrt{ 6 } }$$

Evaluate

$$-\frac{3\sqrt{2}}{2}-\sqrt{3}\approx -3.853371151$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{3}}{6-3\sqrt{6}} by multiplying numerator and denominator by 6+3\sqrt{6}.

$$\frac{3\sqrt{3}\left(6+3\sqrt{6}\right)}{\left(6-3\sqrt{6}\right)\left(6+3\sqrt{6}\right)}$$

Consider \left(6-3\sqrt{6}\right)\left(6+3\sqrt{6}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{3\sqrt{3}\left(6+3\sqrt{6}\right)}{6^{2}-\left(-3\sqrt{6}\right)^{2}}$$

Calculate 6 to the power of 2 and get 36.

$$\frac{3\sqrt{3}\left(6+3\sqrt{6}\right)}{36-\left(-3\sqrt{6}\right)^{2}}$$

Expand \left(-3\sqrt{6}\right)^{2}.

$$\frac{3\sqrt{3}\left(6+3\sqrt{6}\right)}{36-\left(-3\right)^{2}\left(\sqrt{6}\right)^{2}}$$

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

$$\frac{3\sqrt{3}\left(6+3\sqrt{6}\right)}{36-9\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{3\sqrt{3}\left(6+3\sqrt{6}\right)}{36-9\times 6}$$

Multiply 9 and 6 to get 54.

$$\frac{3\sqrt{3}\left(6+3\sqrt{6}\right)}{36-54}$$

Subtract 54 from 36 to get -18.

$$\frac{3\sqrt{3}\left(6+3\sqrt{6}\right)}{-18}$$

Divide 3\sqrt{3}\left(6+3\sqrt{6}\right) by -18 to get -\frac{1}{6}\sqrt{3}\left(6+3\sqrt{6}\right).

$$-\frac{1}{6}\sqrt{3}\left(6+3\sqrt{6}\right)$$

Use the distributive property to multiply -\frac{1}{6}\sqrt{3} by 6+3\sqrt{6}.

$$-\frac{1}{6}\sqrt{3}\times 6-\frac{1}{6}\sqrt{3}\times 3\sqrt{6}$$

Cancel out 6 and 6.

$$-\sqrt{3}-\frac{1}{6}\sqrt{3}\times 3\sqrt{6}$$

Factor 6=3\times 2. Rewrite the square root of the product \sqrt{3\times 2} as the product of square roots \sqrt{3}\sqrt{2}.

$$-\sqrt{3}-\frac{1}{6}\sqrt{3}\times 3\sqrt{3}\sqrt{2}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$-\sqrt{3}-\frac{1}{6}\times 3\times 3\sqrt{2}$$

Express -\frac{1}{6}\times 3 as a single fraction.

$$-\sqrt{3}+\frac{-3}{6}\times 3\sqrt{2}$$

Reduce the fraction \frac{-3}{6} to lowest terms by extracting and canceling out 3.

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

Express -\frac{1}{2}\times 3 as a single fraction.

$$-\sqrt{3}+\frac{-3}{2}\sqrt{2}$$

Fraction \frac{-3}{2} can be rewritten as -\frac{3}{2} by extracting the negative sign.

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