Solve 3/4-sqrt{2}=

$$\frac { 3 } { 4 - \sqrt { 2 } } =$$

Evaluate

$$\frac{3\left(\sqrt{2}+4\right)}{14}\approx 1.160188621$$

Solution Steps

Rationalize the denominator of \frac{3}{4-\sqrt{2}} by multiplying numerator and denominator by 4+\sqrt{2}.

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

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}.

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

Square 4. Square \sqrt{2}.

$$\frac{3\left(4+\sqrt{2}\right)}{16-2}$$

Subtract 2 from 16 to get 14.

$$\frac{3\left(4+\sqrt{2}\right)}{14}$$

Use the distributive property to multiply 3 by 4+\sqrt{2}.

$$\frac{12+3\sqrt{2}}{14}$$