Solve 5/2+sqrt{10}

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

Evaluate

$$\frac{5\left(\sqrt{10}-2\right)}{6}\approx 0.968564717$$

Solution Steps

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

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

Consider \left(2+\sqrt{10}\right)\left(2-\sqrt{10}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square 2. Square \sqrt{10}.

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

Subtract 10 from 4 to get -6.

$$\frac{5\left(2-\sqrt{10}\right)}{-6}$$

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

$$\frac{10-5\sqrt{10}}{-6}$$