Solve frac{3sqrt{10}}{2+sqrt{10}}

$$\frac{ 3 \sqrt{ 10 } }{ 2+ \sqrt{ 10 } }$$

Evaluate

$$5-\sqrt{10}\approx 1.83772234$$

Solution Steps

Rationalize the denominator of \frac{3\sqrt{10}}{2+\sqrt{10}} by multiplying numerator and denominator by 2-\sqrt{10}.

$$\frac{3\sqrt{10}\left(2-\sqrt{10}\right)}{\left(2+\sqrt{10}\right)\left(2-\sqrt{10}\right)}$$

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}.

$$\frac{3\sqrt{10}\left(2-\sqrt{10}\right)}{2^{2}-\left(\sqrt{10}\right)^{2}}$$

Square 2. Square \sqrt{10}.

$$\frac{3\sqrt{10}\left(2-\sqrt{10}\right)}{4-10}$$

Subtract 10 from 4 to get -6.

$$\frac{3\sqrt{10}\left(2-\sqrt{10}\right)}{-6}$$

Use the distributive property to multiply 3\sqrt{10} by 2-\sqrt{10}.

$$\frac{6\sqrt{10}-3\left(\sqrt{10}\right)^{2}}{-6}$$

The square of \sqrt{10} is 10.

$$\frac{6\sqrt{10}-3\times 10}{-6}$$

Multiply -3 and 10 to get -30.

$$\frac{6\sqrt{10}-30}{-6}$$

Divide each term of 6\sqrt{10}-30 by -6 to get -\sqrt{10}+5.

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