Solve 4/2-sqrt{2}-4/sqrt{2}

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

Evaluate

$$4$$

Solution Steps

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

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

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

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

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

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

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

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

The square of \sqrt{2} is 2.

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

Divide 4\sqrt{2} by 2 to get 2\sqrt{2}.

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

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

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

Subtract 2\sqrt{2} from 2\sqrt{2} to get 0.

$$4$$

Factor

$$2^{2}$$