Solve 4/5+sqrt{5}

$$\frac { 4 } { 5 + \sqrt { 5 } }$$

Evaluate

$$-\frac{\sqrt{5}}{5}+1\approx 0.552786405$$

Solution Steps

Rationalize the denominator of \frac{4}{5+\sqrt{5}} by multiplying numerator and denominator by 5-\sqrt{5}.

$$\frac{4\left(5-\sqrt{5}\right)}{\left(5+\sqrt{5}\right)\left(5-\sqrt{5}\right)}$$

Consider \left(5+\sqrt{5}\right)\left(5-\sqrt{5}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 5. Square \sqrt{5}.

$$\frac{4\left(5-\sqrt{5}\right)}{25-5}$$

Subtract 5 from 25 to get 20.

$$\frac{4\left(5-\sqrt{5}\right)}{20}$$

Divide 4\left(5-\sqrt{5}\right) by 20 to get \frac{1}{5}\left(5-\sqrt{5}\right).

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

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

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

Cancel out 5 and 5.

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

Multiply \frac{1}{5} and -1 to get -\frac{1}{5}.

$$1-\frac{1}{5}\sqrt{5}$$