Solve frac{2(3-sqrt{2})}{(3+sqrt{2})*(3-sqrt{2})}

$$\frac { 2 ( 3 - \sqrt { 2 } ) } { ( 3 + \sqrt { 2 } ) \cdot ( 3 - \sqrt { 2 } ) }$$

Evaluate

$$\frac{6-2\sqrt{2}}{7}\approx 0.453081839$$

Solution Steps

Cancel out -\sqrt{2}+3 in both numerator and denominator.

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

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

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

Consider \left(\sqrt{2}+3\right)\left(\sqrt{2}-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{2}. Square 3.

$$\frac{2\left(\sqrt{2}-3\right)}{2-9}$$

Subtract 9 from 2 to get -7.

$$\frac{2\left(\sqrt{2}-3\right)}{-7}$$

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

$$\frac{2\sqrt{2}-6}{-7}$$