Solve sqrt{12}-sqrt{125}-2sqrt{1/3}-sqrt{1/8}+sqrt{8}

$$\sqrt { 12 } - \sqrt { 125 } - 2 \sqrt { \frac { 1 } { 3 } } - \sqrt { \frac { 1 } { 8 } } + \sqrt { 8 }$$

Evaluate

$$\frac{4\sqrt{3}}{3}+\frac{7\sqrt{2}}{4}-5\sqrt{5}\approx -6.396065077$$

Short Solution Steps

Factor 12=2^{2}\times 3. Rewrite the square root of the product \sqrt{2^{2}\times 3} as the product of square roots \sqrt{2^{2}}\sqrt{3}. Take the square root of 2^{2}.

$$2\sqrt{3}-\sqrt{125}-2\sqrt{\frac{1}{3}}-\sqrt{\frac{1}{8}}+\sqrt{8}$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$2\sqrt{3}-5\sqrt{5}-2\sqrt{\frac{1}{3}}-\sqrt{\frac{1}{8}}+\sqrt{8}$$

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

$$2\sqrt{3}-5\sqrt{5}-2\times \frac{\sqrt{1}}{\sqrt{3}}-\sqrt{\frac{1}{8}}+\sqrt{8}$$

Calculate the square root of 1 and get 1.

$$2\sqrt{3}-5\sqrt{5}-2\times \frac{1}{\sqrt{3}}-\sqrt{\frac{1}{8}}+\sqrt{8}$$

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

$$2\sqrt{3}-5\sqrt{5}-2\times \frac{\sqrt{3}}{\left(\sqrt{3}\right)^{2}}-\sqrt{\frac{1}{8}}+\sqrt{8}$$

The square of \sqrt{3} is 3.

$$2\sqrt{3}-5\sqrt{5}-2\times \frac{\sqrt{3}}{3}-\sqrt{\frac{1}{8}}+\sqrt{8}$$

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

$$2\sqrt{3}-5\sqrt{5}+\frac{-2\sqrt{3}}{3}-\sqrt{\frac{1}{8}}+\sqrt{8}$$

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

$$2\sqrt{3}-5\sqrt{5}+\frac{-2\sqrt{3}}{3}-\frac{\sqrt{1}}{\sqrt{8}}+\sqrt{8}$$

Calculate the square root of 1 and get 1.

$$2\sqrt{3}-5\sqrt{5}+\frac{-2\sqrt{3}}{3}-\frac{1}{\sqrt{8}}+\sqrt{8}$$

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}.

$$2\sqrt{3}-5\sqrt{5}+\frac{-2\sqrt{3}}{3}-\frac{1}{2\sqrt{2}}+\sqrt{8}$$

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

$$2\sqrt{3}-5\sqrt{5}+\frac{-2\sqrt{3}}{3}-\frac{\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}+\sqrt{8}$$

The square of \sqrt{2} is 2.

$$2\sqrt{3}-5\sqrt{5}+\frac{-2\sqrt{3}}{3}-\frac{\sqrt{2}}{2\times 2}+\sqrt{8}$$

Multiply 2 and 2 to get 4.

$$2\sqrt{3}-5\sqrt{5}+\frac{-2\sqrt{3}}{3}-\frac{\sqrt{2}}{4}+\sqrt{8}$$

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}.

$$2\sqrt{3}-5\sqrt{5}+\frac{-2\sqrt{3}}{3}-\frac{\sqrt{2}}{4}+2\sqrt{2}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 2\sqrt{3}-5\sqrt{5}+2\sqrt{2} times \frac{3}{3}.

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

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

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

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

$$\frac{6\sqrt{3}-15\sqrt{5}+6\sqrt{2}-2\sqrt{3}}{3}-\frac{\sqrt{2}}{4}$$

Do the calculations in 6\sqrt{3}-15\sqrt{5}+6\sqrt{2}-2\sqrt{3}.

$$\frac{4\sqrt{3}-15\sqrt{5}+6\sqrt{2}}{3}-\frac{\sqrt{2}}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3 and 4 is 12. Multiply \frac{4\sqrt{3}-15\sqrt{5}+6\sqrt{2}}{3} times \frac{4}{4}. Multiply \frac{\sqrt{2}}{4} times \frac{3}{3}.

$$\frac{4\left(4\sqrt{3}-15\sqrt{5}+6\sqrt{2}\right)}{12}-\frac{3\sqrt{2}}{12}$$

Since \frac{4\left(4\sqrt{3}-15\sqrt{5}+6\sqrt{2}\right)}{12} and \frac{3\sqrt{2}}{12} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\left(4\sqrt{3}-15\sqrt{5}+6\sqrt{2}\right)-3\sqrt{2}}{12}$$

Do the multiplications in 4\left(4\sqrt{3}-15\sqrt{5}+6\sqrt{2}\right)-3\sqrt{2}.

$$\frac{16\sqrt{3}-60\sqrt{5}+24\sqrt{2}-3\sqrt{2}}{12}$$

Do the calculations in 16\sqrt{3}-60\sqrt{5}+24\sqrt{2}-3\sqrt{2}.

$$\frac{16\sqrt{3}-60\sqrt{5}+21\sqrt{2}}{12}$$