Solve 2sqrt{11/3}-3sqrt{1/37}+sqrt{75}

$$2 \sqrt { 1 \frac { 1 } { 3 } } - 3 \sqrt { \frac { 1 } { 37 } } + \sqrt { 75 }$$

Evaluate

$$\frac{19\sqrt{3}}{3}-\frac{3\sqrt{37}}{37}\approx 10.476458153$$

Short Solution Steps

Multiply 1 and 3 to get 3.

$$2\sqrt{\frac{3+1}{3}}-3\sqrt{\frac{1}{37}}+\sqrt{75}$$

Add 3 and 1 to get 4.

$$2\sqrt{\frac{4}{3}}-3\sqrt{\frac{1}{37}}+\sqrt{75}$$

Rewrite the square root of the division \sqrt{\frac{4}{3}} as the division of square roots \frac{\sqrt{4}}{\sqrt{3}}.

$$2\times \frac{\sqrt{4}}{\sqrt{3}}-3\sqrt{\frac{1}{37}}+\sqrt{75}$$

Calculate the square root of 4 and get 2.

$$2\times \frac{2}{\sqrt{3}}-3\sqrt{\frac{1}{37}}+\sqrt{75}$$

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

$$2\times \frac{2\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-3\sqrt{\frac{1}{37}}+\sqrt{75}$$

The square of \sqrt{3} is 3.

$$2\times \frac{2\sqrt{3}}{3}-3\sqrt{\frac{1}{37}}+\sqrt{75}$$

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

$$\frac{2\times 2\sqrt{3}}{3}-3\sqrt{\frac{1}{37}}+\sqrt{75}$$

Rewrite the square root of the division \sqrt{\frac{1}{37}} as the division of square roots \frac{\sqrt{1}}{\sqrt{37}}.

$$\frac{2\times 2\sqrt{3}}{3}-3\times \frac{\sqrt{1}}{\sqrt{37}}+\sqrt{75}$$

Calculate the square root of 1 and get 1.

$$\frac{2\times 2\sqrt{3}}{3}-3\times \frac{1}{\sqrt{37}}+\sqrt{75}$$

Rationalize the denominator of \frac{1}{\sqrt{37}} by multiplying numerator and denominator by \sqrt{37}.

$$\frac{2\times 2\sqrt{3}}{3}-3\times \frac{\sqrt{37}}{\left(\sqrt{37}\right)^{2}}+\sqrt{75}$$

The square of \sqrt{37} is 37.

$$\frac{2\times 2\sqrt{3}}{3}-3\times \frac{\sqrt{37}}{37}+\sqrt{75}$$

Express -3\times \frac{\sqrt{37}}{37} as a single fraction.

$$\frac{2\times 2\sqrt{3}}{3}+\frac{-3\sqrt{37}}{37}+\sqrt{75}$$

Factor 75=5^{2}\times 3. Rewrite the square root of the product \sqrt{5^{2}\times 3} as the product of square roots \sqrt{5^{2}}\sqrt{3}. Take the square root of 5^{2}.

$$\frac{2\times 2\sqrt{3}}{3}+\frac{-3\sqrt{37}}{37}+5\sqrt{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 37 is 111. Multiply \frac{2\times 2\sqrt{3}}{3} times \frac{37}{37}. Multiply \frac{-3\sqrt{37}}{37} times \frac{3}{3}.

$$\frac{37\times 2\times 2\sqrt{3}}{111}+\frac{3\left(-1\right)\times 3\sqrt{37}}{111}+5\sqrt{3}$$

Since \frac{37\times 2\times 2\sqrt{3}}{111} and \frac{3\left(-1\right)\times 3\sqrt{37}}{111} have the same denominator, add them by adding their numerators.

$$\frac{37\times 2\times 2\sqrt{3}+3\left(-1\right)\times 3\sqrt{37}}{111}+5\sqrt{3}$$

Do the multiplications in 37\times 2\times 2\sqrt{3}+3\left(-1\right)\times 3\sqrt{37}.

$$\frac{148\sqrt{3}-9\sqrt{37}}{111}+5\sqrt{3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5\sqrt{3} times \frac{111}{111}.

$$\frac{148\sqrt{3}-9\sqrt{37}}{111}+\frac{111\times 5\sqrt{3}}{111}$$

Since \frac{148\sqrt{3}-9\sqrt{37}}{111} and \frac{111\times 5\sqrt{3}}{111} have the same denominator, add them by adding their numerators.

$$\frac{148\sqrt{3}-9\sqrt{37}+111\times 5\sqrt{3}}{111}$$

Do the multiplications in 148\sqrt{3}-9\sqrt{37}+111\times 5\sqrt{3}.

$$\frac{148\sqrt{3}-9\sqrt{37}+555\sqrt{3}}{111}$$

Do the calculations in 148\sqrt{3}-9\sqrt{37}+555\sqrt{3}.

$$\frac{703\sqrt{3}-9\sqrt{37}}{111}$$