Solve 1/sqrt{2}+2/sqrt{8}+3/sqrt{18}

$$\frac { 1 } { \sqrt { 2 } } + \frac { 2 } { \sqrt { 8 } } + \frac { 3 } { \sqrt { 18 } }$$

Evaluate

$$\frac{3\sqrt{2}}{2}\approx 2.121320344$$

Solution Steps

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

$$\frac{\sqrt{2}}{\left(\sqrt{2}\right)^{2}}+\frac{2}{\sqrt{8}}+\frac{3}{\sqrt{18}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{2}+\frac{2}{\sqrt{8}}+\frac{3}{\sqrt{18}}$$

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}.

$$\frac{\sqrt{2}}{2}+\frac{2}{2\sqrt{2}}+\frac{3}{\sqrt{18}}$$

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

$$\frac{\sqrt{2}}{2}+\frac{2\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}+\frac{3}{\sqrt{18}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{2}+\frac{2\sqrt{2}}{2\times 2}+\frac{3}{\sqrt{18}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}+\frac{3}{\sqrt{18}}$$

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

$$\sqrt{2}+\frac{3}{\sqrt{18}}$$

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}.

$$\sqrt{2}+\frac{3}{3\sqrt{2}}$$

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

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

The square of \sqrt{2} is 2.

$$\sqrt{2}+\frac{3\sqrt{2}}{3\times 2}$$

Cancel out 3 in both numerator and denominator.

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

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

$$\frac{3}{2}\sqrt{2}$$