Solve sqrt{(9/5)^2+(3/5)^2}

$$\sqrt { ( \frac { 9 } { 5 } ) ^ { 2 } + ( \frac { 3 } { 5 } ) ^ { 2 } }$$

Evaluate

$$\frac{3\sqrt{10}}{5}\approx 1.897366596$$

Solution Steps

Calculate \frac{9}{5} to the power of 2 and get \frac{81}{25}.

$$\sqrt{\frac{81}{25}+\left(\frac{3}{5}\right)^{2}}$$

Calculate \frac{3}{5} to the power of 2 and get \frac{9}{25}.

$$\sqrt{\frac{81}{25}+\frac{9}{25}}$$

Since \frac{81}{25} and \frac{9}{25} have the same denominator, add them by adding their numerators.

$$\sqrt{\frac{81+9}{25}}$$

Add 81 and 9 to get 90.

$$\sqrt{\frac{90}{25}}$$

Reduce the fraction \frac{90}{25} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{18}{5}}$$

Rewrite the square root of the division \sqrt{\frac{18}{5}} as the division of square roots \frac{\sqrt{18}}{\sqrt{5}}.

$$\frac{\sqrt{18}}{\sqrt{5}}$$

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}.

$$\frac{3\sqrt{2}}{\sqrt{5}}$$

Rationalize the denominator of \frac{3\sqrt{2}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

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

The square of \sqrt{5} is 5.

$$\frac{3\sqrt{2}\sqrt{5}}{5}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{3\sqrt{10}}{5}$$