Solve frac{800*frac{sqrt{2}}{2}}{frac{sqrt{6}-sqrt{2}}{4}}

$$\frac { 800 \times \frac { \sqrt { 2 } } { 2 } } { \frac { \sqrt { 6 } - \sqrt { 2 } } { 4 } }$$

Evaluate

$$800\left(\sqrt{3}+1\right)\approx 2185.640646055$$

Solution Steps

Cancel out 2, the greatest common factor in 800 and 2.

$$\frac{400\sqrt{2}}{\frac{\sqrt{6}-\sqrt{2}}{4}}$$

Divide 400\sqrt{2} by \frac{\sqrt{6}-\sqrt{2}}{4} by multiplying 400\sqrt{2} by the reciprocal of \frac{\sqrt{6}-\sqrt{2}}{4}.

$$\frac{400\sqrt{2}\times 4}{\sqrt{6}-\sqrt{2}}$$

Rationalize the denominator of \frac{400\sqrt{2}\times 4}{\sqrt{6}-\sqrt{2}} by multiplying numerator and denominator by \sqrt{6}+\sqrt{2}.

$$\frac{400\sqrt{2}\times 4\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}-\sqrt{2}\right)\left(\sqrt{6}+\sqrt{2}\right)}$$

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

$$\frac{400\sqrt{2}\times 4\left(\sqrt{6}+\sqrt{2}\right)}{\left(\sqrt{6}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square \sqrt{6}. Square \sqrt{2}.

$$\frac{400\sqrt{2}\times 4\left(\sqrt{6}+\sqrt{2}\right)}{6-2}$$

Subtract 2 from 6 to get 4.

$$\frac{400\sqrt{2}\times 4\left(\sqrt{6}+\sqrt{2}\right)}{4}$$

Multiply 400 and 4 to get 1600.

$$\frac{1600\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)}{4}$$

Divide 1600\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right) by 4 to get 400\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right).

$$400\sqrt{2}\left(\sqrt{6}+\sqrt{2}\right)$$

Use the distributive property to multiply 400\sqrt{2} by \sqrt{6}+\sqrt{2}.

$$400\sqrt{2}\sqrt{6}+400\left(\sqrt{2}\right)^{2}$$

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}.

$$400\sqrt{2}\sqrt{2}\sqrt{3}+400\left(\sqrt{2}\right)^{2}$$

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

$$400\times 2\sqrt{3}+400\left(\sqrt{2}\right)^{2}$$

Multiply 400 and 2 to get 800.

$$800\sqrt{3}+400\left(\sqrt{2}\right)^{2}$$

The square of \sqrt{2} is 2.

$$800\sqrt{3}+400\times 2$$

Multiply 400 and 2 to get 800.

$$800\sqrt{3}+800$$