Solve sqrt{(3/4sqrt{2})^2+(5/4sqrt{2})^2}

$$\sqrt { ( \frac { 3 } { 4 } \sqrt { 2 } ) ^ { 2 } + ( \frac { 5 } { 4 } \sqrt { 2 } ) ^ { 2 } }$$

Evaluate

$$\frac{\sqrt{17}}{2}\approx 2.061552813$$

Solution Steps

Expand \left(\frac{3}{4}\sqrt{2}\right)^{2}.

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

Calculate \frac{3}{4} to the power of 2 and get \frac{9}{16}.

$$\sqrt{\frac{9}{16}\left(\sqrt{2}\right)^{2}+\left(\frac{5}{4}\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{\frac{9}{16}\times 2+\left(\frac{5}{4}\sqrt{2}\right)^{2}}$$

Multiply \frac{9}{16} and 2 to get \frac{9}{8}.

$$\sqrt{\frac{9}{8}+\left(\frac{5}{4}\sqrt{2}\right)^{2}}$$

Expand \left(\frac{5}{4}\sqrt{2}\right)^{2}.

$$\sqrt{\frac{9}{8}+\left(\frac{5}{4}\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate \frac{5}{4} to the power of 2 and get \frac{25}{16}.

$$\sqrt{\frac{9}{8}+\frac{25}{16}\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\sqrt{\frac{9}{8}+\frac{25}{16}\times 2}$$

Multiply \frac{25}{16} and 2 to get \frac{25}{8}.

$$\sqrt{\frac{9}{8}+\frac{25}{8}}$$

Add \frac{9}{8} and \frac{25}{8} to get \frac{17}{4}.

$$\sqrt{\frac{17}{4}}$$

Rewrite the square root of the division \sqrt{\frac{17}{4}} as the division of square roots \frac{\sqrt{17}}{\sqrt{4}}.

$$\frac{\sqrt{17}}{\sqrt{4}}$$

Calculate the square root of 4 and get 2.

$$\frac{\sqrt{17}}{2}$$