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

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

Evaluate

$$1$$

Solution Steps

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

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

Cancel out 2 in both numerator and denominator.

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

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

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

Consider \left(-2\sqrt{2}+4\right)\left(-2\sqrt{2}-4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate -2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

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

Calculate 4 to the power of 2 and get 16.

$$\frac{2\left(-\sqrt{2}+2\right)\left(-2\sqrt{2}-4\right)}{8-16}$$

Subtract 16 from 8 to get -8.

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

Divide 2\left(-\sqrt{2}+2\right)\left(-2\sqrt{2}-4\right) by -8 to get -\frac{1}{4}\left(-\sqrt{2}+2\right)\left(-2\sqrt{2}-4\right).

$$-\frac{1}{4}\left(-\sqrt{2}+2\right)\left(-2\sqrt{2}-4\right)$$

Use the distributive property to multiply -\frac{1}{4} by -\sqrt{2}+2.

$$\left(-\frac{1}{4}\left(-1\right)\sqrt{2}-\frac{1}{4}\times 2\right)\left(-2\sqrt{2}-4\right)$$

Multiply -\frac{1}{4} and -1 to get \frac{1}{4}.

$$\left(\frac{1}{4}\sqrt{2}-\frac{1}{4}\times 2\right)\left(-2\sqrt{2}-4\right)$$

Express -\frac{1}{4}\times 2 as a single fraction.

$$\left(\frac{1}{4}\sqrt{2}+\frac{-2}{4}\right)\left(-2\sqrt{2}-4\right)$$

Reduce the fraction \frac{-2}{4} to lowest terms by extracting and canceling out 2.

$$\left(\frac{1}{4}\sqrt{2}-\frac{1}{2}\right)\left(-2\sqrt{2}-4\right)$$

Apply the distributive property by multiplying each term of \frac{1}{4}\sqrt{2}-\frac{1}{2} by each term of -2\sqrt{2}-4.

$$\frac{1}{4}\sqrt{2}\left(-2\right)\sqrt{2}+\frac{1}{4}\sqrt{2}\left(-4\right)-\frac{1}{2}\left(-2\right)\sqrt{2}-\frac{1}{2}\left(-4\right)$$

Multiply \sqrt{2} and \sqrt{2} to get 2.

$$\frac{1}{4}\times 2\left(-2\right)+\frac{1}{4}\sqrt{2}\left(-4\right)-\frac{1}{2}\left(-2\right)\sqrt{2}-\frac{1}{2}\left(-4\right)$$

Multiply \frac{1}{4} and 2 to get \frac{2}{4}.

$$\frac{2}{4}\left(-2\right)+\frac{1}{4}\sqrt{2}\left(-4\right)-\frac{1}{2}\left(-2\right)\sqrt{2}-\frac{1}{2}\left(-4\right)$$

Reduce the fraction \frac{2}{4} to lowest terms by extracting and canceling out 2.

$$\frac{1}{2}\left(-2\right)+\frac{1}{4}\sqrt{2}\left(-4\right)-\frac{1}{2}\left(-2\right)\sqrt{2}-\frac{1}{2}\left(-4\right)$$

Multiply \frac{1}{2} and -2 to get \frac{-2}{2}.

$$\frac{-2}{2}+\frac{1}{4}\sqrt{2}\left(-4\right)-\frac{1}{2}\left(-2\right)\sqrt{2}-\frac{1}{2}\left(-4\right)$$

Divide -2 by 2 to get -1.

$$-1+\frac{1}{4}\sqrt{2}\left(-4\right)-\frac{1}{2}\left(-2\right)\sqrt{2}-\frac{1}{2}\left(-4\right)$$

Multiply \frac{1}{4} and -4 to get \frac{-4}{4}.

$$-1+\frac{-4}{4}\sqrt{2}-\frac{1}{2}\left(-2\right)\sqrt{2}-\frac{1}{2}\left(-4\right)$$

Divide -4 by 4 to get -1.

$$-1-\sqrt{2}-\frac{1}{2}\left(-2\right)\sqrt{2}-\frac{1}{2}\left(-4\right)$$

Express -\frac{1}{2}\left(-2\right) as a single fraction.

$$-1-\sqrt{2}+\frac{-\left(-2\right)}{2}\sqrt{2}-\frac{1}{2}\left(-4\right)$$

Multiply -1 and -2 to get 2.

$$-1-\sqrt{2}+\frac{2}{2}\sqrt{2}-\frac{1}{2}\left(-4\right)$$

Divide 2 by 2 to get 1.

$$-1-\sqrt{2}+1\sqrt{2}-\frac{1}{2}\left(-4\right)$$

Combine -\sqrt{2} and 1\sqrt{2} to get 0.

$$-1-\frac{1}{2}\left(-4\right)$$

Express -\frac{1}{2}\left(-4\right) as a single fraction.

$$-1+\frac{-\left(-4\right)}{2}$$

Multiply -1 and -4 to get 4.

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

Divide 4 by 2 to get 2.

$$-1+2$$

Add -1 and 2 to get 1.

$$1$$

Factor

$$1$$