Solve -{[(1/sqrt{2})-(-1/sqrt{5})]+(-sqrt{4})^3+2(sqrt{16}-1/2)}div(3/4)

$$- \{ [ ( \frac { 1 } { \sqrt { 2 } } ) - ( - \frac { 1 } { \sqrt { 5 } } ) ] + ( - \sqrt { 4 } ) ^ { 3 } + 2 ( \sqrt { 16 } - \frac { 1 } { 2 } ) \} \div ( \frac { 3 } { 4 } )$$

Evaluate

$$-\frac{2\sqrt{2}}{3}-\frac{4\sqrt{5}}{15}+\frac{4}{3}\approx -0.205760502$$

Short Solution Steps

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

$$\frac{-\left(\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}-\left(-\frac{1}{\sqrt{5}}\right)+\left(-\sqrt{4}\right)^{3}+2\left(\sqrt{16}-\frac{1}{2}\right)\right)}{\frac{3}{4}}$$

The square of \sqrt{2} is 2.

$$\frac{-\left(\frac{\sqrt{2}}{2}-\left(-\frac{1}{\sqrt{5}}\right)+\left(-\sqrt{4}\right)^{3}+2\left(\sqrt{16}-\frac{1}{2}\right)\right)}{\frac{3}{4}}$$

Rationalize the denominator of \frac{1}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{-\left(\frac{\sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\right)+\left(-\sqrt{4}\right)^{3}+2\left(\sqrt{16}-\frac{1}{2}\right)\right)}{\frac{3}{4}}$$

The square of \sqrt{5} is 5.

$$\frac{-\left(\frac{\sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{5}\right)+\left(-\sqrt{4}\right)^{3}+2\left(\sqrt{16}-\frac{1}{2}\right)\right)}{\frac{3}{4}}$$

Calculate the square root of 4 and get 2.

$$\frac{-\left(\frac{\sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{5}\right)+\left(-2\right)^{3}+2\left(\sqrt{16}-\frac{1}{2}\right)\right)}{\frac{3}{4}}$$

Calculate -2 to the power of 3 and get -8.

$$\frac{-\left(\frac{\sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{5}\right)-8+2\left(\sqrt{16}-\frac{1}{2}\right)\right)}{\frac{3}{4}}$$

Calculate the square root of 16 and get 4.

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

Subtract \frac{1}{2} from 4 to get \frac{7}{2}.

$$\frac{-\left(\frac{\sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{5}\right)-8+2\times \frac{7}{2}\right)}{\frac{3}{4}}$$

Multiply 2 and \frac{7}{2} to get 7.

$$\frac{-\left(\frac{\sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{5}\right)-8+7\right)}{\frac{3}{4}}$$

Add -8 and 7 to get -1.

$$\frac{-\left(\frac{\sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{5}\right)-1\right)}{\frac{3}{4}}$$

To find the opposite of \frac{\sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{5}\right)-1, find the opposite of each term.

$$\frac{-\left(\frac{\sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{5}\right)\right)+1}{\frac{3}{4}}$$

Divide -\left(\frac{\sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{5}\right)\right)+1 by \frac{3}{4} by multiplying -\left(\frac{\sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{5}\right)\right)+1 by the reciprocal of \frac{3}{4}.

$$\frac{\left(-\left(\frac{\sqrt{2}}{2}-\left(-\frac{\sqrt{5}}{5}\right)\right)+1\right)\times 4}{3}$$

Multiply -1 and -1 to get 1.

$$\frac{\left(-\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{5}}{5}\right)+1\right)\times 4}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2 and 5 is 10. Multiply \frac{\sqrt{2}}{2} times \frac{5}{5}. Multiply \frac{\sqrt{5}}{5} times \frac{2}{2}.

$$\frac{\left(-\left(\frac{5\sqrt{2}}{10}+\frac{2\sqrt{5}}{10}\right)+1\right)\times 4}{3}$$

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

$$\frac{\left(-\frac{5\sqrt{2}+2\sqrt{5}}{10}+1\right)\times 4}{3}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{10}{10}.

$$\frac{\left(-\frac{5\sqrt{2}+2\sqrt{5}}{10}+\frac{10}{10}\right)\times 4}{3}$$

Since -\frac{5\sqrt{2}+2\sqrt{5}}{10} and \frac{10}{10} have the same denominator, add them by adding their numerators.

$$\frac{\frac{-\left(5\sqrt{2}+2\sqrt{5}\right)+10}{10}\times 4}{3}$$

Do the multiplications in -\left(5\sqrt{2}+2\sqrt{5}\right)+10.

$$\frac{\frac{-5\sqrt{2}-2\sqrt{5}+10}{10}\times 4}{3}$$

Express \frac{-5\sqrt{2}-2\sqrt{5}+10}{10}\times 4 as a single fraction.

$$\frac{\frac{\left(-5\sqrt{2}-2\sqrt{5}+10\right)\times 4}{10}}{3}$$

Express \frac{\frac{\left(-5\sqrt{2}-2\sqrt{5}+10\right)\times 4}{10}}{3} as a single fraction.

$$\frac{\left(-5\sqrt{2}-2\sqrt{5}+10\right)\times 4}{10\times 3}$$

Cancel out 2 in both numerator and denominator.

$$\frac{2\left(-5\sqrt{2}-2\sqrt{5}+10\right)}{3\times 5}$$

Multiply 3 and 5 to get 15.

$$\frac{2\left(-5\sqrt{2}-2\sqrt{5}+10\right)}{15}$$

Use the distributive property to multiply 2 by -5\sqrt{2}-2\sqrt{5}+10.

$$\frac{-10\sqrt{2}-4\sqrt{5}+20}{15}$$