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

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

Evaluate

$$\sqrt{2}+\frac{1}{2}\approx 1.914213562$$

Solution Steps

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}.

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

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

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

Calculate the square root of 1 and get 1.

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

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

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

The square of \sqrt{2} is 2.

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

Cancel out 2 and 2.

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

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

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

Factor

$$\frac{2 \sqrt{2} + 1}{2} = 1.9142135623730951$$