Solve 4/1-sqrt{2/2}

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

Evaluate

$$4\sqrt{2}+8\approx 13.656854249$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{2}{2}.

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

Since \frac{2}{2} and \frac{\sqrt{2}}{2} have the same denominator, subtract them by subtracting their numerators.

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

Divide 4 by \frac{2-\sqrt{2}}{2} by multiplying 4 by the reciprocal of \frac{2-\sqrt{2}}{2}.

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

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

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

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}.

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

Multiply 4 and 2 to get 8.

$$\frac{8\left(2+\sqrt{2}\right)}{2}$$

Divide 8\left(2+\sqrt{2}\right) by 2 to get 4\left(2+\sqrt{2}\right).

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

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

$$8+4\sqrt{2}$$