Solve 3/1-sqrt{2}

$$\frac{ 3 }{ 1- \sqrt{ 2 } }$$

Evaluate

$$-3\sqrt{2}-3\approx -7.242640687$$

Solution Steps

Rationalize the denominator of \frac{3}{1-\sqrt{2}} by multiplying numerator and denominator by 1+\sqrt{2}.

$$\frac{3\left(1+\sqrt{2}\right)}{\left(1-\sqrt{2}\right)\left(1+\sqrt{2}\right)}$$

Consider \left(1-\sqrt{2}\right)\left(1+\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{3\left(1+\sqrt{2}\right)}{1^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 1. Square \sqrt{2}.

$$\frac{3\left(1+\sqrt{2}\right)}{1-2}$$

Subtract 2 from 1 to get -1.

$$\frac{3\left(1+\sqrt{2}\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-3\left(1+\sqrt{2}\right)$$

Use the distributive property to multiply -3 by 1+\sqrt{2}.

$$-3-3\sqrt{2}$$