Solve 8/sqrt{4-sqrt{2}}

$$\frac{ 8 }{ \sqrt{ 4 } - \sqrt{ 2 } }$$

Evaluate

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

Solution Steps

Calculate the square root of 4 and get 2.

$$\frac{8}{2-\sqrt{2}}$$

Rationalize the denominator of \frac{8}{2-\sqrt{2}} by multiplying numerator and denominator by 2+\sqrt{2}.

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

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}.

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

Square 2. Square \sqrt{2}.

$$\frac{8\left(2+\sqrt{2}\right)}{4-2}$$

Subtract 2 from 4 to get 2.

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

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}.

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