Solve 2/sqrt{12-sqrt{8}}

$$\frac { 2 } { \sqrt { 12 } - \sqrt { 8 } }$$

Evaluate

$$\sqrt{2}+\sqrt{3}\approx 3.14626437$$

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}.

$$\frac{2}{2\sqrt{3}-\sqrt{8}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

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

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

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

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{3} is 3.

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

Multiply 4 and 3 to get 12.

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

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

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

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

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

$$\frac{2\left(2\sqrt{3}+2\sqrt{2}\right)}{12-8}$$

Subtract 8 from 12 to get 4.

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

Divide 2\left(2\sqrt{3}+2\sqrt{2}\right) by 4 to get \frac{1}{2}\left(2\sqrt{3}+2\sqrt{2}\right).

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

Use the distributive property to multiply \frac{1}{2} by 2\sqrt{3}+2\sqrt{2}.

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

Cancel out 2 and 2.

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

Cancel out 2 and 2.

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