Solve frac{4-sqrt{2}}{2sqrt{2}-2}

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

Evaluate

$$\frac{3\sqrt{2}}{2}+1\approx 3.121320344$$

Solution Steps

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

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

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

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

Expand \left(2\sqrt{2}\right)^{2}.

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

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

Calculate 2 to the power of 2 and get 4.

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

Subtract 4 from 8 to get 4.

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

Apply the distributive property by multiplying each term of 4-\sqrt{2} by each term of 2\sqrt{2}+2.

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

The square of \sqrt{2} is 2.

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

Multiply -2 and 2 to get -4.

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

Subtract 4 from 8 to get 4.

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

Combine 8\sqrt{2} and -2\sqrt{2} to get 6\sqrt{2}.

$$\frac{6\sqrt{2}+4}{4}$$