Solve frac{frac{sqrt{3}}{3}*50}{frac{3+sqrt{3}}{3}*sqrt{2}}

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

Evaluate

$$\frac{25\sqrt{2}\left(\sqrt{3}-1\right)}{2}\approx 12.940952255$$

Solution Steps

Express 50\times \frac{\sqrt{3}}{3} as a single fraction.

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

Express \frac{3+\sqrt{3}}{3}\sqrt{2} as a single fraction.

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

Divide \frac{50\sqrt{3}}{3} by \frac{\left(3+\sqrt{3}\right)\sqrt{2}}{3} by multiplying \frac{50\sqrt{3}}{3} by the reciprocal of \frac{\left(3+\sqrt{3}\right)\sqrt{2}}{3}.

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

Cancel out 3 in both numerator and denominator.

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

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

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

The square of \sqrt{2} is 2.

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

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

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

Cancel out 2 in both numerator and denominator.

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

Rationalize the denominator of \frac{25\sqrt{6}}{\sqrt{3}+3} by multiplying numerator and denominator by \sqrt{3}-3.

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

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

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

Square \sqrt{3}. Square 3.

$$\frac{25\sqrt{6}\left(\sqrt{3}-3\right)}{3-9}$$

Subtract 9 from 3 to get -6.

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

Use the distributive property to multiply 25\sqrt{6} by \sqrt{3}-3.

$$\frac{25\sqrt{6}\sqrt{3}-75\sqrt{6}}{-6}$$

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}.

$$\frac{25\sqrt{3}\sqrt{2}\sqrt{3}-75\sqrt{6}}{-6}$$

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

$$\frac{25\times 3\sqrt{2}-75\sqrt{6}}{-6}$$

Multiply 25 and 3 to get 75.

$$\frac{75\sqrt{2}-75\sqrt{6}}{-6}$$