Solve sqrt{6}div(1/sqrt{3}+sqrt{2})+sqrt{50}

$$\sqrt { 6 } \div ( \frac { 1 } { \sqrt { 3 } } + \sqrt { 2 } ) + \sqrt { 50 }$$

Evaluate

$$\frac{2\left(3\sqrt{3}+11\sqrt{2}\right)}{5}\approx 8.301000644$$

Short Solution Steps

Rationalize the denominator of \frac{1}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{2} times \frac{3}{3}.

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

Since \frac{\sqrt{3}}{3} and \frac{3\sqrt{2}}{3} have the same denominator, add them by adding their numerators.

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

Divide \sqrt{6} by \frac{\sqrt{3}+3\sqrt{2}}{3} by multiplying \sqrt{6} by the reciprocal of \frac{\sqrt{3}+3\sqrt{2}}{3}.

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

Rationalize the denominator of \frac{\sqrt{6}\times 3}{\sqrt{3}+3\sqrt{2}} by multiplying numerator and denominator by \sqrt{3}-3\sqrt{2}.

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

Consider \left(\sqrt{3}+3\sqrt{2}\right)\left(\sqrt{3}-3\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}.

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

The square of \sqrt{3} is 3.

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

Expand \left(3\sqrt{2}\right)^{2}.

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

Calculate 3 to the power of 2 and get 9.

$$\frac{\sqrt{6}\times 3\left(\sqrt{3}-3\sqrt{2}\right)}{3-9\left(\sqrt{2}\right)^{2}}+\sqrt{50}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{6}\times 3\left(\sqrt{3}-3\sqrt{2}\right)}{3-9\times 2}+\sqrt{50}$$

Multiply 9 and 2 to get 18.

$$\frac{\sqrt{6}\times 3\left(\sqrt{3}-3\sqrt{2}\right)}{3-18}+\sqrt{50}$$

Subtract 18 from 3 to get -15.

$$\frac{\sqrt{6}\times 3\left(\sqrt{3}-3\sqrt{2}\right)}{-15}+\sqrt{50}$$

Divide \sqrt{6}\times 3\left(\sqrt{3}-3\sqrt{2}\right) by -15 to get \sqrt{6}\left(-\frac{1}{5}\right)\left(\sqrt{3}-3\sqrt{2}\right).

$$\sqrt{6}\left(-\frac{1}{5}\right)\left(\sqrt{3}-3\sqrt{2}\right)+\sqrt{50}$$

Factor 50=5^{2}\times 2. Rewrite the square root of the product \sqrt{5^{2}\times 2} as the product of square roots \sqrt{5^{2}}\sqrt{2}. Take the square root of 5^{2}.

$$\sqrt{6}\left(-\frac{1}{5}\right)\left(\sqrt{3}-3\sqrt{2}\right)+5\sqrt{2}$$

Use the distributive property to multiply \sqrt{6}\left(-\frac{1}{5}\right) by \sqrt{3}-3\sqrt{2}.

$$\sqrt{6}\left(-\frac{1}{5}\right)\sqrt{3}+\sqrt{6}\left(-\frac{1}{5}\right)\left(-3\right)\sqrt{2}+5\sqrt{2}$$

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}.

$$\sqrt{3}\sqrt{2}\left(-\frac{1}{5}\right)\sqrt{3}+\sqrt{6}\left(-\frac{1}{5}\right)\left(-3\right)\sqrt{2}+5\sqrt{2}$$

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

$$3\left(-\frac{1}{5}\right)\sqrt{2}+\sqrt{6}\left(-\frac{1}{5}\right)\left(-3\right)\sqrt{2}+5\sqrt{2}$$

Express 3\left(-\frac{1}{5}\right) as a single fraction.

$$\frac{3\left(-1\right)}{5}\sqrt{2}+\sqrt{6}\left(-\frac{1}{5}\right)\left(-3\right)\sqrt{2}+5\sqrt{2}$$

Multiply 3 and -1 to get -3.

$$\frac{-3}{5}\sqrt{2}+\sqrt{6}\left(-\frac{1}{5}\right)\left(-3\right)\sqrt{2}+5\sqrt{2}$$

Fraction \frac{-3}{5} can be rewritten as -\frac{3}{5} by extracting the negative sign.

$$-\frac{3}{5}\sqrt{2}+\sqrt{6}\left(-\frac{1}{5}\right)\left(-3\right)\sqrt{2}+5\sqrt{2}$$

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}.

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

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

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

Express 2\left(-\frac{1}{5}\right) as a single fraction.

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

Multiply 2 and -1 to get -2.

$$-\frac{3}{5}\sqrt{2}+\frac{-2}{5}\left(-3\right)\sqrt{3}+5\sqrt{2}$$

Fraction \frac{-2}{5} can be rewritten as -\frac{2}{5} by extracting the negative sign.

$$-\frac{3}{5}\sqrt{2}-\frac{2}{5}\left(-3\right)\sqrt{3}+5\sqrt{2}$$

Express -\frac{2}{5}\left(-3\right) as a single fraction.

$$-\frac{3}{5}\sqrt{2}+\frac{-2\left(-3\right)}{5}\sqrt{3}+5\sqrt{2}$$

Multiply -2 and -3 to get 6.

$$-\frac{3}{5}\sqrt{2}+\frac{6}{5}\sqrt{3}+5\sqrt{2}$$

Combine -\frac{3}{5}\sqrt{2} and 5\sqrt{2} to get \frac{22}{5}\sqrt{2}.

$$\frac{22}{5}\sqrt{2}+\frac{6}{5}\sqrt{3}$$