Solve (frac{sqrt{5}-5}{sqrt{5}})^2

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

Evaluate

$$6-2\sqrt{5}\approx 1.527864045$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

To raise \frac{\left(\sqrt{5}-5\right)\sqrt{5}}{5} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-5\right)^{2}.

$$\frac{\left(\left(\sqrt{5}\right)^{2}-10\sqrt{5}+25\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(5-10\sqrt{5}+25\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

Add 5 and 25 to get 30.

$$\frac{\left(30-10\sqrt{5}\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(30-10\sqrt{5}\right)\times 5}{5^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(30-10\sqrt{5}\right)\times 5}{25}$$

Divide \left(30-10\sqrt{5}\right)\times 5 by 25 to get \left(30-10\sqrt{5}\right)\times \frac{1}{5}.

$$\left(30-10\sqrt{5}\right)\times \frac{1}{5}$$

Use the distributive property to multiply 30-10\sqrt{5} by \frac{1}{5}.

$$6-2\sqrt{5}$$

Expand

$$6-2\sqrt{5}$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

To raise \frac{\left(\sqrt{5}-5\right)\sqrt{5}}{5} to a power, raise both numerator and denominator to the power and then divide.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{5}-5\right)^{2}.

$$\frac{\left(\left(\sqrt{5}\right)^{2}-10\sqrt{5}+25\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(5-10\sqrt{5}+25\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

Add 5 and 25 to get 30.

$$\frac{\left(30-10\sqrt{5}\right)\left(\sqrt{5}\right)^{2}}{5^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(30-10\sqrt{5}\right)\times 5}{5^{2}}$$

Calculate 5 to the power of 2 and get 25.

$$\frac{\left(30-10\sqrt{5}\right)\times 5}{25}$$

Divide \left(30-10\sqrt{5}\right)\times 5 by 25 to get \left(30-10\sqrt{5}\right)\times \frac{1}{5}.

$$\left(30-10\sqrt{5}\right)\times \frac{1}{5}$$

Use the distributive property to multiply 30-10\sqrt{5} by \frac{1}{5}.

$$6-2\sqrt{5}$$