Solve sqrt{6}(4/sqrt{2}+6/sqrt{3})-(2/sqrt{78}+15/2sqrt{75})*6

$$\sqrt { 6 } ( \frac { 4 } { \sqrt { 2 } } + \frac { 6 } { \sqrt { 3 } } ) - ( \frac { 2 } { \sqrt { 78 } } + \frac { 15 } { 2 \sqrt { 75 } } ) \cdot 6$$

Evaluate

$$-\frac{2\sqrt{78}}{13}+\sqrt{3}+6\sqrt{2}\approx 8.858599741$$

Solution Steps

Rationalize the denominator of \frac{4}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\sqrt{6}\left(\frac{4\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\frac{6}{\sqrt{3}}\right)-6\left(\frac{2}{\sqrt{78}}+\frac{15}{2\sqrt{75}}\right)$$

The square of \sqrt{2} is 2.

$$\sqrt{6}\left(\frac{4\sqrt{2}}{2}+\frac{6}{\sqrt{3}}\right)-6\left(\frac{2}{\sqrt{78}}+\frac{15}{2\sqrt{75}}\right)$$

Divide 4\sqrt{2} by 2 to get 2\sqrt{2}.

$$\sqrt{6}\left(2\sqrt{2}+\frac{6}{\sqrt{3}}\right)-6\left(\frac{2}{\sqrt{78}}+\frac{15}{2\sqrt{75}}\right)$$

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

$$\sqrt{6}\left(2\sqrt{2}+\frac{6\sqrt{3}}{\left(\sqrt{3}\right)^{2}}\right)-6\left(\frac{2}{\sqrt{78}}+\frac{15}{2\sqrt{75}}\right)$$

The square of \sqrt{3} is 3.

$$\sqrt{6}\left(2\sqrt{2}+\frac{6\sqrt{3}}{3}\right)-6\left(\frac{2}{\sqrt{78}}+\frac{15}{2\sqrt{75}}\right)$$

Divide 6\sqrt{3} by 3 to get 2\sqrt{3}.

$$\sqrt{6}\left(2\sqrt{2}+2\sqrt{3}\right)-6\left(\frac{2}{\sqrt{78}}+\frac{15}{2\sqrt{75}}\right)$$

Rationalize the denominator of \frac{2}{\sqrt{78}} by multiplying numerator and denominator by \sqrt{78}.

$$\sqrt{6}\left(2\sqrt{2}+2\sqrt{3}\right)-6\left(\frac{2\sqrt{78}}{\left(\sqrt{78}\right)^{2}}+\frac{15}{2\sqrt{75}}\right)$$

The square of \sqrt{78} is 78.

$$\sqrt{6}\left(2\sqrt{2}+2\sqrt{3}\right)-6\left(\frac{2\sqrt{78}}{78}+\frac{15}{2\sqrt{75}}\right)$$

Divide 2\sqrt{78} by 78 to get \frac{1}{39}\sqrt{78}.

$$\sqrt{6}\left(2\sqrt{2}+2\sqrt{3}\right)-6\left(\frac{1}{39}\sqrt{78}+\frac{15}{2\sqrt{75}}\right)$$

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}.

$$\sqrt{6}\left(2\sqrt{2}+2\sqrt{3}\right)-6\left(\frac{1}{39}\sqrt{78}+\frac{15}{2\times 5\sqrt{3}}\right)$$

Multiply 2 and 5 to get 10.

$$\sqrt{6}\left(2\sqrt{2}+2\sqrt{3}\right)-6\left(\frac{1}{39}\sqrt{78}+\frac{15}{10\sqrt{3}}\right)$$

Rationalize the denominator of \frac{15}{10\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\sqrt{6}\left(2\sqrt{2}+2\sqrt{3}\right)-6\left(\frac{1}{39}\sqrt{78}+\frac{15\sqrt{3}}{10\left(\sqrt{3}\right)^{2}}\right)$$

The square of \sqrt{3} is 3.

$$\sqrt{6}\left(2\sqrt{2}+2\sqrt{3}\right)-6\left(\frac{1}{39}\sqrt{78}+\frac{15\sqrt{3}}{10\times 3}\right)$$

Cancel out 3\times 5 in both numerator and denominator.

$$\sqrt{6}\left(2\sqrt{2}+2\sqrt{3}\right)-6\left(\frac{1}{39}\sqrt{78}+\frac{\sqrt{3}}{2}\right)$$

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

$$2\sqrt{6}\sqrt{2}+2\sqrt{6}\sqrt{3}-6\left(\frac{1}{39}\sqrt{78}+\frac{\sqrt{3}}{2}\right)$$

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}.

$$2\sqrt{2}\sqrt{3}\sqrt{2}+2\sqrt{6}\sqrt{3}-6\left(\frac{1}{39}\sqrt{78}+\frac{\sqrt{3}}{2}\right)$$

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

$$2\times 2\sqrt{3}+2\sqrt{6}\sqrt{3}-6\left(\frac{1}{39}\sqrt{78}+\frac{\sqrt{3}}{2}\right)$$

Multiply 2 and 2 to get 4.

$$4\sqrt{3}+2\sqrt{6}\sqrt{3}-6\left(\frac{1}{39}\sqrt{78}+\frac{\sqrt{3}}{2}\right)$$

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}.

$$4\sqrt{3}+2\sqrt{3}\sqrt{2}\sqrt{3}-6\left(\frac{1}{39}\sqrt{78}+\frac{\sqrt{3}}{2}\right)$$

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

$$4\sqrt{3}+2\times 3\sqrt{2}-6\left(\frac{1}{39}\sqrt{78}+\frac{\sqrt{3}}{2}\right)$$

Multiply 2 and 3 to get 6.

$$4\sqrt{3}+6\sqrt{2}-6\left(\frac{1}{39}\sqrt{78}+\frac{\sqrt{3}}{2}\right)$$

Use the distributive property to multiply -6 by \frac{1}{39}\sqrt{78}+\frac{\sqrt{3}}{2}.

$$4\sqrt{3}+6\sqrt{2}-6\times \frac{1}{39}\sqrt{78}-6\times \frac{\sqrt{3}}{2}$$

Multiply -6 and \frac{1}{39} to get \frac{-6}{39}.

$$4\sqrt{3}+6\sqrt{2}+\frac{-6}{39}\sqrt{78}-6\times \frac{\sqrt{3}}{2}$$

Reduce the fraction \frac{-6}{39} to lowest terms by extracting and canceling out 3.

$$4\sqrt{3}+6\sqrt{2}-\frac{2}{13}\sqrt{78}-6\times \frac{\sqrt{3}}{2}$$

Cancel out 2, the greatest common factor in 6 and 2.

$$4\sqrt{3}+6\sqrt{2}-\frac{2}{13}\sqrt{78}-3\sqrt{3}$$

Combine 4\sqrt{3} and -3\sqrt{3} to get \sqrt{3}.

$$\sqrt{3}+6\sqrt{2}-\frac{2}{13}\sqrt{78}$$