Solve frac{sqrt{3}+sqrt{2}}{sqrt{22}-sqrt{8}}

$$\frac { \sqrt { 3 } + \sqrt { 2 } } { \sqrt { 22 } - \sqrt { 8 } }$$

Evaluate

$$\frac{\sqrt{66}+2\sqrt{6}+2\sqrt{11}+4}{14}\approx 1.689733391$$

Solution Steps

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}.

$$\frac{\sqrt{3}+\sqrt{2}}{\sqrt{22}-2\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{3}+\sqrt{2}}{\sqrt{22}-2\sqrt{2}} by multiplying numerator and denominator by \sqrt{22}+2\sqrt{2}.

$$\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{22}+2\sqrt{2}\right)}{\left(\sqrt{22}-2\sqrt{2}\right)\left(\sqrt{22}+2\sqrt{2}\right)}$$

Consider \left(\sqrt{22}-2\sqrt{2}\right)\left(\sqrt{22}+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}.

$$\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{22}+2\sqrt{2}\right)}{\left(\sqrt{22}\right)^{2}-\left(-2\sqrt{2}\right)^{2}}$$

The square of \sqrt{22} is 22.

$$\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{22}+2\sqrt{2}\right)}{22-\left(-2\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{22}+2\sqrt{2}\right)}{22-\left(-2\right)^{2}\left(\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{22}+2\sqrt{2}\right)}{22-4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{22}+2\sqrt{2}\right)}{22-4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{22}+2\sqrt{2}\right)}{22-8}$$

Subtract 8 from 22 to get 14.

$$\frac{\left(\sqrt{3}+\sqrt{2}\right)\left(\sqrt{22}+2\sqrt{2}\right)}{14}$$

Apply the distributive property by multiplying each term of \sqrt{3}+\sqrt{2} by each term of \sqrt{22}+2\sqrt{2}.

$$\frac{\sqrt{3}\sqrt{22}+2\sqrt{3}\sqrt{2}+\sqrt{2}\sqrt{22}+2\left(\sqrt{2}\right)^{2}}{14}$$

To multiply \sqrt{3} and \sqrt{22}, multiply the numbers under the square root.

$$\frac{\sqrt{66}+2\sqrt{3}\sqrt{2}+\sqrt{2}\sqrt{22}+2\left(\sqrt{2}\right)^{2}}{14}$$

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{66}+2\sqrt{6}+\sqrt{2}\sqrt{22}+2\left(\sqrt{2}\right)^{2}}{14}$$

Factor 22=2\times 11. Rewrite the square root of the product \sqrt{2\times 11} as the product of square roots \sqrt{2}\sqrt{11}.

$$\frac{\sqrt{66}+2\sqrt{6}+\sqrt{2}\sqrt{2}\sqrt{11}+2\left(\sqrt{2}\right)^{2}}{14}$$

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

$$\frac{\sqrt{66}+2\sqrt{6}+2\sqrt{11}+2\left(\sqrt{2}\right)^{2}}{14}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{66}+2\sqrt{6}+2\sqrt{11}+2\times 2}{14}$$

Multiply 2 and 2 to get 4.

$$\frac{\sqrt{66}+2\sqrt{6}+2\sqrt{11}+4}{14}$$