Solve 27/1-sqrt{10}

$$\frac{ 27 }{ 1- \sqrt{ 10 } }$$

Evaluate

$$-3\sqrt{10}-3\approx -12.486832981$$

Solution Steps

Rationalize the denominator of \frac{27}{1-\sqrt{10}} by multiplying numerator and denominator by 1+\sqrt{10}.

$$\frac{27\left(1+\sqrt{10}\right)}{\left(1-\sqrt{10}\right)\left(1+\sqrt{10}\right)}$$

Consider \left(1-\sqrt{10}\right)\left(1+\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}.

$$\frac{27\left(1+\sqrt{10}\right)}{1^{2}-\left(\sqrt{10}\right)^{2}}$$

Square 1. Square \sqrt{10}.

$$\frac{27\left(1+\sqrt{10}\right)}{1-10}$$

Subtract 10 from 1 to get -9.

$$\frac{27\left(1+\sqrt{10}\right)}{-9}$$

Divide 27\left(1+\sqrt{10}\right) by -9 to get -3\left(1+\sqrt{10}\right).

$$-3\left(1+\sqrt{10}\right)$$

Use the distributive property to multiply -3 by 1+\sqrt{10}.

$$-3-3\sqrt{10}$$