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

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

Evaluate

$$\frac{50\left(\sqrt{6}+3\sqrt{2}\right)}{3}\approx 111.535507165$$

Solution Steps

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

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

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}.

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

Multiply 25 and 2 to get 50.

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

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

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

The square of \sqrt{3} is 3.

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

Use the distributive property to multiply 50 by \sqrt{6}+\sqrt{2}.

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

Use the distributive property to multiply 50\sqrt{6}+50\sqrt{2} by \sqrt{3}.

$$\frac{50\sqrt{6}\sqrt{3}+50\sqrt{2}\sqrt{3}}{3}$$

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

$$\frac{50\sqrt{3}\sqrt{2}\sqrt{3}+50\sqrt{2}\sqrt{3}}{3}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{50\times 3\sqrt{2}+50\sqrt{2}\sqrt{3}}{3}$$

Multiply 50 and 3 to get 150.

$$\frac{150\sqrt{2}+50\sqrt{2}\sqrt{3}}{3}$$

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

$$\frac{150\sqrt{2}+50\sqrt{6}}{3}$$