Solve frac{2sqrt{2}}{3-2sqrt{6}}

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

Evaluate

$$-\frac{2\sqrt{2}}{5}-\frac{8\sqrt{3}}{15}\approx -1.489445856$$

Solution Steps

Rationalize the denominator of \frac{2\sqrt{2}}{3-2\sqrt{6}} by multiplying numerator and denominator by 3+2\sqrt{6}.

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

Consider \left(3-2\sqrt{6}\right)\left(3+2\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}.

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

Calculate 3 to the power of 2 and get 9.

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

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

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

Calculate -2 to the power of 2 and get 4.

$$\frac{2\sqrt{2}\left(3+2\sqrt{6}\right)}{9-4\left(\sqrt{6}\right)^{2}}$$

The square of \sqrt{6} is 6.

$$\frac{2\sqrt{2}\left(3+2\sqrt{6}\right)}{9-4\times 6}$$

Multiply 4 and 6 to get 24.

$$\frac{2\sqrt{2}\left(3+2\sqrt{6}\right)}{9-24}$$

Subtract 24 from 9 to get -15.

$$\frac{2\sqrt{2}\left(3+2\sqrt{6}\right)}{-15}$$

Use the distributive property to multiply 2\sqrt{2} by 3+2\sqrt{6}.

$$\frac{6\sqrt{2}+4\sqrt{2}\sqrt{6}}{-15}$$

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

$$\frac{6\sqrt{2}+4\sqrt{2}\sqrt{2}\sqrt{3}}{-15}$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{6\sqrt{2}+4\times 2\sqrt{3}}{-15}$$

Multiply 4 and 2 to get 8.

$$\frac{6\sqrt{2}+8\sqrt{3}}{-15}$$