Solve frac{5-2sqrt{5}}{3sqrt{5}}*frac{3sqrt{5}}{3sqrt{5}}

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

Evaluate

$$\frac{\sqrt{5}-2}{3}\approx 0.078689326$$

Solution Steps

Divide 3\sqrt{5} by 3\sqrt{5} to get 1.

$$\frac{5-2\sqrt{5}}{3\sqrt{5}}\times 1$$

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

$$\frac{\left(5-2\sqrt{5}\right)\sqrt{5}}{3\left(\sqrt{5}\right)^{2}}\times 1$$

The square of \sqrt{5} is 5.

$$\frac{\left(5-2\sqrt{5}\right)\sqrt{5}}{3\times 5}\times 1$$

Multiply 3 and 5 to get 15.

$$\frac{\left(5-2\sqrt{5}\right)\sqrt{5}}{15}\times 1$$

Express \frac{\left(5-2\sqrt{5}\right)\sqrt{5}}{15}\times 1 as a single fraction.

$$\frac{\left(5-2\sqrt{5}\right)\sqrt{5}}{15}$$

Use the distributive property to multiply 5-2\sqrt{5} by \sqrt{5}.

$$\frac{5\sqrt{5}-2\left(\sqrt{5}\right)^{2}}{15}$$

The square of \sqrt{5} is 5.

$$\frac{5\sqrt{5}-2\times 5}{15}$$

Multiply -2 and 5 to get -10.

$$\frac{5\sqrt{5}-10}{15}$$

Factor

$$\frac{\sqrt{5} - 2}{3} = 0.07868932583326327$$