Solve 5sqrt{8}-sqrt{25/2})-8sqrt{1/3}-(4/sqrt{3}+6sqrt{1+1/3})

$$5 \sqrt { 8 } - \sqrt { \frac { 25 } { 2 } } ) - 8 \sqrt { \frac { 1 } { 3 } } - ( \frac { 4 } { \sqrt { 3 } } + 6 \sqrt { 1 + \frac { 1 } { 3 } } )$$

Evaluate

$$\frac{15\sqrt{2}}{2}\approx 10.606601718$$

Short Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$5\times 2\sqrt{2}-\sqrt{\frac{25}{2}}$$

Multiply 5 and 2 to get 10.

$$10\sqrt{2}-\sqrt{\frac{25}{2}}$$

Rewrite the square root of the division \sqrt{\frac{25}{2}} as the division of square roots \frac{\sqrt{25}}{\sqrt{2}}.

$$10\sqrt{2}-\frac{\sqrt{25}}{\sqrt{2}}$$

Calculate the square root of 25 and get 5.

$$10\sqrt{2}-\frac{5}{\sqrt{2}}$$

Rationalize the denominator of \frac{5}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$10\sqrt{2}-\frac{5\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$10\sqrt{2}-\frac{5\sqrt{2}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 10\sqrt{2} times \frac{2}{2}.

$$\frac{2\times 10\sqrt{2}}{2}-\frac{5\sqrt{2}}{2}$$

Since \frac{2\times 10\sqrt{2}}{2} and \frac{5\sqrt{2}}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2\times 10\sqrt{2}-5\sqrt{2}}{2}$$

Do the multiplications in 2\times 10\sqrt{2}-5\sqrt{2}.

$$\frac{20\sqrt{2}-5\sqrt{2}}{2}$$

Do the calculations in 20\sqrt{2}-5\sqrt{2}.

$$\frac{15\sqrt{2}}{2}$$