Solve sqrt{1/2}-sqrt{8}+1/sqrt{2}-1/sqrt{8}

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

Evaluate

$$-\frac{5\sqrt{2}}{4}\approx -1.767766953$$

Solution Steps

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{2} is 2.

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

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}.

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

Combine \frac{\sqrt{2}}{2} and -2\sqrt{2} to get -\frac{3}{2}\sqrt{2}.

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

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

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

The square of \sqrt{2} is 2.

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

Combine -\frac{3}{2}\sqrt{2} and \frac{\sqrt{2}}{2} to get -\sqrt{2}.

$$-\sqrt{2}-\frac{1}{\sqrt{8}}$$

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}.

$$-\sqrt{2}-\frac{1}{2\sqrt{2}}$$

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

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

Combine -\sqrt{2} and -\frac{\sqrt{2}}{4} to get -\frac{5}{4}\sqrt{2}.

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