Solve 30/sqrt{2}+30/sqrt{6}+30/sqrt{15}-30/sqrt{10}-sqrt{2}(5sqrt{3}-3sqrt{5})

$$\frac { 30 } { \sqrt { 2 } } + \frac { 30 } { \sqrt { 6 } } + \frac { 30 } { \sqrt { 15 } } - \frac { 30 } { \sqrt { 10 } } - \sqrt { 2 } ( 5 \sqrt { 3 } - 3 \sqrt { 5 } )$$

Evaluate

$$2\sqrt{15}+15\sqrt{2}\approx 28.959170128$$

Solution Steps

Rationalize the denominator of \frac{30}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{30\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\frac{30}{\sqrt{6}}+\frac{30}{\sqrt{15}}-\frac{30}{\sqrt{10}}-\sqrt{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

The square of \sqrt{2} is 2.

$$\frac{30\sqrt{2}}{2}+\frac{30}{\sqrt{6}}+\frac{30}{\sqrt{15}}-\frac{30}{\sqrt{10}}-\sqrt{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

Divide 30\sqrt{2} by 2 to get 15\sqrt{2}.

$$15\sqrt{2}+\frac{30}{\sqrt{6}}+\frac{30}{\sqrt{15}}-\frac{30}{\sqrt{10}}-\sqrt{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

Rationalize the denominator of \frac{30}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$15\sqrt{2}+\frac{30\sqrt{6}}{\left(\sqrt{6}\right)^{2}}+\frac{30}{\sqrt{15}}-\frac{30}{\sqrt{10}}-\sqrt{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

The square of \sqrt{6} is 6.

$$15\sqrt{2}+\frac{30\sqrt{6}}{6}+\frac{30}{\sqrt{15}}-\frac{30}{\sqrt{10}}-\sqrt{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

Divide 30\sqrt{6} by 6 to get 5\sqrt{6}.

$$15\sqrt{2}+5\sqrt{6}+\frac{30}{\sqrt{15}}-\frac{30}{\sqrt{10}}-\sqrt{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

Rationalize the denominator of \frac{30}{\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$15\sqrt{2}+5\sqrt{6}+\frac{30\sqrt{15}}{\left(\sqrt{15}\right)^{2}}-\frac{30}{\sqrt{10}}-\sqrt{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

The square of \sqrt{15} is 15.

$$15\sqrt{2}+5\sqrt{6}+\frac{30\sqrt{15}}{15}-\frac{30}{\sqrt{10}}-\sqrt{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

Divide 30\sqrt{15} by 15 to get 2\sqrt{15}.

$$15\sqrt{2}+5\sqrt{6}+2\sqrt{15}-\frac{30}{\sqrt{10}}-\sqrt{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

Rationalize the denominator of \frac{30}{\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$15\sqrt{2}+5\sqrt{6}+2\sqrt{15}-\frac{30\sqrt{10}}{\left(\sqrt{10}\right)^{2}}-\sqrt{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

The square of \sqrt{10} is 10.

$$15\sqrt{2}+5\sqrt{6}+2\sqrt{15}-\frac{30\sqrt{10}}{10}-\sqrt{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

Divide 30\sqrt{10} by 10 to get 3\sqrt{10}.

$$15\sqrt{2}+5\sqrt{6}+2\sqrt{15}-3\sqrt{10}-\sqrt{2}\left(5\sqrt{3}-3\sqrt{5}\right)$$

Use the distributive property to multiply \sqrt{2} by 5\sqrt{3}-3\sqrt{5}.

$$15\sqrt{2}+5\sqrt{6}+2\sqrt{15}-3\sqrt{10}-\left(5\sqrt{2}\sqrt{3}-3\sqrt{2}\sqrt{5}\right)$$

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

$$15\sqrt{2}+5\sqrt{6}+2\sqrt{15}-3\sqrt{10}-\left(5\sqrt{6}-3\sqrt{2}\sqrt{5}\right)$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$15\sqrt{2}+5\sqrt{6}+2\sqrt{15}-3\sqrt{10}-\left(5\sqrt{6}-3\sqrt{10}\right)$$

To find the opposite of 5\sqrt{6}-3\sqrt{10}, find the opposite of each term.

$$15\sqrt{2}+5\sqrt{6}+2\sqrt{15}-3\sqrt{10}-5\sqrt{6}-\left(-3\sqrt{10}\right)$$

Combine 5\sqrt{6} and -5\sqrt{6} to get 0.

$$15\sqrt{2}+2\sqrt{15}-3\sqrt{10}-\left(-3\sqrt{10}\right)$$

The opposite of -3\sqrt{10} is 3\sqrt{10}.

$$15\sqrt{2}+2\sqrt{15}-3\sqrt{10}+3\sqrt{10}$$

Combine -3\sqrt{10} and 3\sqrt{10} to get 0.

$$15\sqrt{2}+2\sqrt{15}$$