Solve sqrt{18}-sqrt{8}+sqrt{1/8}

$$\sqrt { 18 } - \sqrt { 8 } + \sqrt { \frac { 1 } { 8 } }$$

Evaluate

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

Solution Steps

Factor 18=3^{2}\times 2. Rewrite the square root of the product \sqrt{3^{2}\times 2} as the product of square roots \sqrt{3^{2}}\sqrt{2}. Take the square root of 3^{2}.

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

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}.

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

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

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

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

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

Calculate the square root of 1 and get 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}.

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

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}.

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