Solve frac{100*frac{sqrt{6}+sqrt{2}}{4}}{frac{sqrt{3}}{2}}*frac{sqrt{2}}{2}

$$\frac { 100 \times \frac { \sqrt { 6 } + \sqrt { 2 } } { 4 } } { \frac { \sqrt { 3 } } { 2 } } \times \frac { \sqrt { 2 } } { 2 }$$

Evaluate

$$\frac{50\sqrt{3}}{3}+50\approx 78.867513459$$

Solution Steps

Cancel out 4, the greatest common factor in 100 and 4.

$$\frac{25\left(\sqrt{6}+\sqrt{2}\right)}{\frac{\sqrt{3}}{2}}\times \frac{\sqrt{2}}{2}$$

Divide 25\left(\sqrt{6}+\sqrt{2}\right) by \frac{\sqrt{3}}{2} by multiplying 25\left(\sqrt{6}+\sqrt{2}\right) by the reciprocal of \frac{\sqrt{3}}{2}.

$$\frac{25\left(\sqrt{6}+\sqrt{2}\right)\times 2}{\sqrt{3}}\times \frac{\sqrt{2}}{2}$$

Multiply 25 and 2 to get 50.

$$\frac{50\left(\sqrt{6}+\sqrt{2}\right)}{\sqrt{3}}\times \frac{\sqrt{2}}{2}$$

Rationalize the denominator of \frac{50\left(\sqrt{6}+\sqrt{2}\right)}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{50\left(\sqrt{6}+\sqrt{2}\right)\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\times \frac{\sqrt{2}}{2}$$

The square of \sqrt{3} is 3.

$$\frac{50\left(\sqrt{6}+\sqrt{2}\right)\sqrt{3}}{3}\times \frac{\sqrt{2}}{2}$$

Multiply \frac{50\left(\sqrt{6}+\sqrt{2}\right)\sqrt{3}}{3} times \frac{\sqrt{2}}{2} by multiplying numerator times numerator and denominator times denominator.

$$\frac{50\left(\sqrt{6}+\sqrt{2}\right)\sqrt{3}\sqrt{2}}{3\times 2}$$

Cancel out 2 in both numerator and denominator.

$$\frac{25\sqrt{2}\sqrt{3}\left(\sqrt{2}+\sqrt{6}\right)}{3}$$

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

$$\frac{25\sqrt{6}\left(\sqrt{2}+\sqrt{6}\right)}{3}$$

Use the distributive property to multiply 25\sqrt{6} by \sqrt{2}+\sqrt{6}.

$$\frac{25\sqrt{6}\sqrt{2}+25\left(\sqrt{6}\right)^{2}}{3}$$

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}.

$$\frac{25\sqrt{2}\sqrt{3}\sqrt{2}+25\left(\sqrt{6}\right)^{2}}{3}$$

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

$$\frac{25\times 2\sqrt{3}+25\left(\sqrt{6}\right)^{2}}{3}$$

Multiply 25 and 2 to get 50.

$$\frac{50\sqrt{3}+25\left(\sqrt{6}\right)^{2}}{3}$$

The square of \sqrt{6} is 6.

$$\frac{50\sqrt{3}+25\times 6}{3}$$

Multiply 25 and 6 to get 150.

$$\frac{50\sqrt{3}+150}{3}$$