Solve 7/4-sqrt{2}

$$\frac{ 7 }{ 4- \sqrt{ 2 } }$$

Evaluate

$$\frac{\sqrt{2}}{2}+2\approx 2.707106781$$

Solution Steps

Rationalize the denominator of \frac{7}{4-\sqrt{2}} by multiplying numerator and denominator by 4+\sqrt{2}.

$$\frac{7\left(4+\sqrt{2}\right)}{\left(4-\sqrt{2}\right)\left(4+\sqrt{2}\right)}$$

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}.

$$\frac{7\left(4+\sqrt{2}\right)}{4^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 4. Square \sqrt{2}.

$$\frac{7\left(4+\sqrt{2}\right)}{16-2}$$

Subtract 2 from 16 to get 14.

$$\frac{7\left(4+\sqrt{2}\right)}{14}$$

Divide 7\left(4+\sqrt{2}\right) by 14 to get \frac{1}{2}\left(4+\sqrt{2}\right).

$$\frac{1}{2}\left(4+\sqrt{2}\right)$$

Use the distributive property to multiply \frac{1}{2} by 4+\sqrt{2}.

$$\frac{1}{2}\times 4+\frac{1}{2}\sqrt{2}$$

Multiply \frac{1}{2} and 4 to get \frac{4}{2}.

$$\frac{4}{2}+\frac{1}{2}\sqrt{2}$$

Divide 4 by 2 to get 2.

$$2+\frac{1}{2}\sqrt{2}$$