Solve (sqrt{32}-2sqrt{1/3})-(sqrt{1/8}-sqrt{75})

$$( \sqrt { 32 } - 2 \sqrt { \frac { 1 } { 3 } } ) - ( \sqrt { \frac { 1 } { 8 } } - \sqrt { 75 } )$$

Evaluate

$$\frac{13\sqrt{3}}{3}+\frac{15\sqrt{2}}{4}\approx 12.808854358$$

Short Solution Steps

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$4\sqrt{2}-2\sqrt{\frac{1}{3}}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Rewrite the square root of the division \sqrt{\frac{1}{3}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3}}.

$$4\sqrt{2}-2\times \frac{\sqrt{1}}{\sqrt{3}}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Calculate the square root of 1 and get 1.

$$4\sqrt{2}-2\times \frac{1}{\sqrt{3}}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

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

$$4\sqrt{2}-2\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

The square of \sqrt{3} is 3.

$$4\sqrt{2}-2\times \frac{\sqrt{3}}{3}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

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

$$4\sqrt{2}+\frac{-2\sqrt{3}}{3}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4\sqrt{2} times \frac{3}{3}.

$$\frac{3\times 4\sqrt{2}}{3}+\frac{-2\sqrt{3}}{3}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Since \frac{3\times 4\sqrt{2}}{3} and \frac{-2\sqrt{3}}{3} have the same denominator, add them by adding their numerators.

$$\frac{3\times 4\sqrt{2}-2\sqrt{3}}{3}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Do the multiplications in 3\times 4\sqrt{2}-2\sqrt{3}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)$$

Rewrite the square root of the division \sqrt{\frac{1}{8}} as the division of square roots \frac{\sqrt{1}}{\sqrt{8}}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{\sqrt{1}}{\sqrt{8}}-\sqrt{75}\right)$$

Calculate the square root of 1 and get 1.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{1}{\sqrt{8}}-\sqrt{75}\right)$$

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}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{1}{2\sqrt{2}}-\sqrt{75}\right)$$

Rationalize the denominator of \frac{1}{2\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}-\sqrt{75}\right)$$

The square of \sqrt{2} is 2.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{\sqrt{2}}{2\times 2}-\sqrt{75}\right)$$

Multiply 2 and 2 to get 4.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{\sqrt{2}}{4}-\sqrt{75}\right)$$

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}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{\sqrt{2}}{4}-5\sqrt{3}\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -5\sqrt{3} times \frac{4}{4}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\left(\frac{\sqrt{2}}{4}+\frac{4\left(-5\right)\sqrt{3}}{4}\right)$$

Since \frac{\sqrt{2}}{4} and \frac{4\left(-5\right)\sqrt{3}}{4} have the same denominator, add them by adding their numerators.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\frac{\sqrt{2}+4\left(-5\right)\sqrt{3}}{4}$$

Do the multiplications in \sqrt{2}+4\left(-5\right)\sqrt{3}.

$$\frac{12\sqrt{2}-2\sqrt{3}}{3}-\frac{\sqrt{2}-20\sqrt{3}}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 4 is 12. Multiply \frac{12\sqrt{2}-2\sqrt{3}}{3} times \frac{4}{4}. Multiply \frac{\sqrt{2}-20\sqrt{3}}{4} times \frac{3}{3}.

$$\frac{4\left(12\sqrt{2}-2\sqrt{3}\right)}{12}-\frac{3\left(\sqrt{2}-20\sqrt{3}\right)}{12}$$

Since \frac{4\left(12\sqrt{2}-2\sqrt{3}\right)}{12} and \frac{3\left(\sqrt{2}-20\sqrt{3}\right)}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\left(12\sqrt{2}-2\sqrt{3}\right)-3\left(\sqrt{2}-20\sqrt{3}\right)}{12}$$

Do the multiplications in 4\left(12\sqrt{2}-2\sqrt{3}\right)-3\left(\sqrt{2}-20\sqrt{3}\right).

$$\frac{48\sqrt{2}-8\sqrt{3}-3\sqrt{2}+60\sqrt{3}}{12}$$

Do the calculations in 48\sqrt{2}-8\sqrt{3}-3\sqrt{2}+60\sqrt{3}.

$$\frac{45\sqrt{2}+52\sqrt{3}}{12}$$