Solve 10/2-sqrt{5}*frac{2+sqrt{5}}{2+sqrt{5}}

$$\frac { 10 } { 2 - \sqrt { 5 } } \cdot \frac { 2 + \sqrt { 5 } } { 2 + \sqrt { 5 } }$$

Evaluate

$$-10\sqrt{5}-20\approx -42.360679775$$

Solution Steps

Divide 2+\sqrt{5} by 2+\sqrt{5} to get 1.

$$\frac{10}{2-\sqrt{5}}\times 1$$

Rationalize the denominator of \frac{10}{2-\sqrt{5}} by multiplying numerator and denominator by 2+\sqrt{5}.

$$\frac{10\left(2+\sqrt{5}\right)}{\left(2-\sqrt{5}\right)\left(2+\sqrt{5}\right)}\times 1$$

Consider \left(2-\sqrt{5}\right)\left(2+\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}.

$$\frac{10\left(2+\sqrt{5}\right)}{2^{2}-\left(\sqrt{5}\right)^{2}}\times 1$$

Square 2. Square \sqrt{5}.

$$\frac{10\left(2+\sqrt{5}\right)}{4-5}\times 1$$

Subtract 5 from 4 to get -1.

$$\frac{10\left(2+\sqrt{5}\right)}{-1}\times 1$$

Anything divided by -1 gives its opposite.

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

Use the distributive property to multiply -10 by 2+\sqrt{5}.

$$\left(-20-10\sqrt{5}\right)\times 1$$

Use the distributive property to multiply -20-10\sqrt{5} by 1.

$$-20-10\sqrt{5}$$