Solve sqrt{32}-sqrt{25}-sqrt{1/2}-2sqrt{2/3}

$$\sqrt { 32 } - \sqrt { 25 } - \sqrt { \frac { 1 } { 2 } } - 2 \sqrt { \frac { 2 } { 3 } }$$

Evaluate

$$\frac{7\sqrt{2}}{2}-\frac{2\sqrt{6}}{3}-5\approx -1.683245694$$

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}.

$$4\sqrt{2}-\sqrt{25}-\sqrt{\frac{1}{2}}-2\sqrt{\frac{2}{3}}$$

Calculate the square root of 25 and get 5.

$$4\sqrt{2}-5-\sqrt{\frac{1}{2}}-2\sqrt{\frac{2}{3}}$$

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

$$4\sqrt{2}-5-\frac{\sqrt{1}}{\sqrt{2}}-2\sqrt{\frac{2}{3}}$$

Calculate the square root of 1 and get 1.

$$4\sqrt{2}-5-\frac{1}{\sqrt{2}}-2\sqrt{\frac{2}{3}}$$

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

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

The square of \sqrt{2} is 2.

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

Rewrite the square root of the division \sqrt{\frac{2}{3}} as the division of square roots \frac{\sqrt{2}}{\sqrt{3}}.

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

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

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

The square of \sqrt{3} is 3.

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

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

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

Express -2\times \frac{\sqrt{6}}{3} as a single fraction.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply 4\sqrt{2}-5 times \frac{2}{2}.

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

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

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

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

$$\frac{8\sqrt{2}-10-\sqrt{2}}{2}+\frac{-2\sqrt{6}}{3}$$

Do the calculations in 8\sqrt{2}-10-\sqrt{2}.

$$\frac{7\sqrt{2}-10}{2}+\frac{-2\sqrt{6}}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 3 is 6. Multiply \frac{7\sqrt{2}-10}{2} times \frac{3}{3}. Multiply \frac{-2\sqrt{6}}{3} times \frac{2}{2}.

$$\frac{3\left(7\sqrt{2}-10\right)}{6}+\frac{2\left(-1\right)\times 2\sqrt{6}}{6}$$

Since \frac{3\left(7\sqrt{2}-10\right)}{6} and \frac{2\left(-1\right)\times 2\sqrt{6}}{6} have the same denominator, add them by adding their numerators.

$$\frac{3\left(7\sqrt{2}-10\right)+2\left(-1\right)\times 2\sqrt{6}}{6}$$

Do the multiplications in 3\left(7\sqrt{2}-10\right)+2\left(-1\right)\times 2\sqrt{6}.

$$\frac{21\sqrt{2}-30-4\sqrt{6}}{6}$$