Solve (5-sqrt{5})divsqrt{5}+sqrt{5}(2sqrt{5}-2)

$$( 5 - \sqrt { 5 } ) \div \sqrt { 5 } + \sqrt { 5 } ( 2 \sqrt { 5 } - 2 )$$

Evaluate

$$9-\sqrt{5}\approx 6.763932023$$

Solution Steps

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

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

The square of \sqrt{5} is 5.

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

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

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

The square of \sqrt{5} is 5.

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

Divide each term of 5\sqrt{5}-5 by 5 to get \sqrt{5}-1.

$$\sqrt{5}-1+\sqrt{5}\left(2\sqrt{5}-2\right)$$

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

$$\sqrt{5}-1+2\left(\sqrt{5}\right)^{2}-2\sqrt{5}$$

The square of \sqrt{5} is 5.

$$\sqrt{5}-1+2\times 5-2\sqrt{5}$$

Multiply 2 and 5 to get 10.

$$\sqrt{5}-1+10-2\sqrt{5}$$

Add -1 and 10 to get 9.

$$\sqrt{5}+9-2\sqrt{5}$$

Combine \sqrt{5} and -2\sqrt{5} to get -\sqrt{5}.

$$-\sqrt{5}+9$$

Factor

$$9-\sqrt{5}$$