Solve frac{8sqrt{2}}{2-sqrt{2}}

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

Evaluate

$$8\left(\sqrt{2}+1\right)\approx 19.313708499$$

Solution Steps

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

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

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}.

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

Use the distributive property to multiply 8\sqrt{2} by 2+\sqrt{2}.

$$\frac{16\sqrt{2}+8\left(\sqrt{2}\right)^{2}}{2}$$

The square of \sqrt{2} is 2.

$$\frac{16\sqrt{2}+8\times 2}{2}$$

Multiply 8 and 2 to get 16.

$$\frac{16\sqrt{2}+16}{2}$$