Solve frac{6-sqrt{2}}{4+sqrt{2}}

$$\frac { 6 - \sqrt { 2 } } { 4 + \sqrt { 2 } }$$

Evaluate

$$\frac{13-5\sqrt{2}}{7}\approx 0.846990313$$

Solution Steps

Rationalize the denominator of \frac{6-\sqrt{2}}{4+\sqrt{2}} by multiplying numerator and denominator by 4-\sqrt{2}.

$$\frac{\left(6-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{\left(4+\sqrt{2}\right)\left(4-\sqrt{2}\right)}$$

Consider \left(4+\sqrt{2}\right)\left(4-\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}.

$$\frac{\left(6-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{4^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 4. Square \sqrt{2}.

$$\frac{\left(6-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{16-2}$$

Subtract 2 from 16 to get 14.

$$\frac{\left(6-\sqrt{2}\right)\left(4-\sqrt{2}\right)}{14}$$

Apply the distributive property by multiplying each term of 6-\sqrt{2} by each term of 4-\sqrt{2}.

$$\frac{24-6\sqrt{2}-4\sqrt{2}+\left(\sqrt{2}\right)^{2}}{14}$$

Combine -6\sqrt{2} and -4\sqrt{2} to get -10\sqrt{2}.

$$\frac{24-10\sqrt{2}+\left(\sqrt{2}\right)^{2}}{14}$$

The square of \sqrt{2} is 2.

$$\frac{24-10\sqrt{2}+2}{14}$$

Add 24 and 2 to get 26.

$$\frac{26-10\sqrt{2}}{14}$$