Solve frac{sqrt{2}-1}{2sqrt{2}+1}*frac{sqrt{2}+1}{-sqrt{2}+1}=

$$\frac { \sqrt { 2 } - 1 } { 2 \sqrt { 2 } + 1 } \cdot \frac { \sqrt { 2 } + 1 } { - \sqrt { 2 } + 1 } =$$

Evaluate

$$\frac{-\sqrt{2}-3}{7}\approx -0.630601937$$

Solution Steps

Rationalize the denominator of \frac{\sqrt{2}-1}{2\sqrt{2}+1} by multiplying numerator and denominator by 2\sqrt{2}-1.

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

Consider \left(2\sqrt{2}+1\right)\left(2\sqrt{2}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

The square of \sqrt{2} is 2.

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

Multiply 4 and 2 to get 8.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{8-1^{2}}\times \frac{\sqrt{2}+1}{-\sqrt{2}+1}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{8-1}\times \frac{\sqrt{2}+1}{-\sqrt{2}+1}$$

Subtract 1 from 8 to get 7.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\sqrt{2}+1}{-\sqrt{2}+1}$$

Rationalize the denominator of \frac{\sqrt{2}+1}{-\sqrt{2}+1} by multiplying numerator and denominator by -\sqrt{2}-1.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}{\left(-\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}$$

Consider \left(-\sqrt{2}+1\right)\left(-\sqrt{2}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}{\left(-\sqrt{2}\right)^{2}-1^{2}}$$

Calculate -\sqrt{2} to the power of 2 and get \left(\sqrt{2}\right)^{2}.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}{\left(\sqrt{2}\right)^{2}-1^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}{\left(\sqrt{2}\right)^{2}-1}$$

The square of \sqrt{2} is 2.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}{2-1}$$

Subtract 1 from 2 to get 1.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\times \frac{\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)}{1}$$

Anything divided by one gives itself.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\left(\sqrt{2}+1\right)\left(-\sqrt{2}-1\right)$$

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

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)$$

Express \frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)}{7}\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right) as a single fraction.

$$\frac{\left(\sqrt{2}-1\right)\left(2\sqrt{2}-1\right)\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)}{7}$$

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

$$\frac{\left(2\left(\sqrt{2}\right)^{2}-\sqrt{2}-2\sqrt{2}+1\right)\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)}{7}$$

The square of \sqrt{2} is 2.

$$\frac{\left(2\times 2-\sqrt{2}-2\sqrt{2}+1\right)\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)}{7}$$

Multiply 2 and 2 to get 4.

$$\frac{\left(4-\sqrt{2}-2\sqrt{2}+1\right)\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)}{7}$$

Combine -\sqrt{2} and -2\sqrt{2} to get -3\sqrt{2}.

$$\frac{\left(4-3\sqrt{2}+1\right)\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)}{7}$$

Add 4 and 1 to get 5.

$$\frac{\left(5-3\sqrt{2}\right)\left(\sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1\right)}{7}$$

Apply the distributive property by multiplying each term of 5-3\sqrt{2} by each term of \sqrt{2}\left(-\sqrt{2}\right)-\sqrt{2}-\sqrt{2}-1.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)-5\sqrt{2}+5\left(-\sqrt{2}\right)-5-3\left(-\sqrt{2}\right)\left(\sqrt{2}\right)^{2}+3\left(\sqrt{2}\right)^{2}-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

The square of \sqrt{2} is 2.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)-5\sqrt{2}+5\left(-\sqrt{2}\right)-5-3\left(-\sqrt{2}\right)\times 2+3\left(\sqrt{2}\right)^{2}-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Multiply -3 and 2 to get -6.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)-5\sqrt{2}+5\left(-\sqrt{2}\right)-5-6\left(-\sqrt{2}\right)+3\left(\sqrt{2}\right)^{2}-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Multiply -6 and -1 to get 6.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)-5\sqrt{2}+5\left(-\sqrt{2}\right)-5+6\sqrt{2}+3\left(\sqrt{2}\right)^{2}-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Combine -5\sqrt{2} and 6\sqrt{2} to get \sqrt{2}.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+\sqrt{2}+5\left(-\sqrt{2}\right)-5+3\left(\sqrt{2}\right)^{2}-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

The square of \sqrt{2} is 2.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+\sqrt{2}+5\left(-\sqrt{2}\right)-5+3\times 2-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Multiply 3 and 2 to get 6.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+\sqrt{2}+5\left(-\sqrt{2}\right)-5+6-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Add -5 and 6 to get 1.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+\sqrt{2}+5\left(-\sqrt{2}\right)+1-3\sqrt{2}\left(-\sqrt{2}\right)+3\sqrt{2}}{7}$$

Multiply -3 and -1 to get 3.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+\sqrt{2}+5\left(-\sqrt{2}\right)+1+3\sqrt{2}\sqrt{2}+3\sqrt{2}}{7}$$

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

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+\sqrt{2}+5\left(-\sqrt{2}\right)+1+3\times 2+3\sqrt{2}}{7}$$

Combine \sqrt{2} and 3\sqrt{2} to get 4\sqrt{2}.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+4\sqrt{2}+5\left(-\sqrt{2}\right)+1+3\times 2}{7}$$

Multiply 3 and 2 to get 6.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+4\sqrt{2}+5\left(-\sqrt{2}\right)+1+6}{7}$$

Add 1 and 6 to get 7.

$$\frac{5\sqrt{2}\left(-\sqrt{2}\right)+4\sqrt{2}+5\left(-\sqrt{2}\right)+7}{7}$$

Multiply 5 and -1 to get -5.

$$\frac{-5\sqrt{2}\sqrt{2}+4\sqrt{2}+5\left(-1\right)\sqrt{2}+7}{7}$$

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

$$\frac{-5\times 2+4\sqrt{2}+5\left(-1\right)\sqrt{2}+7}{7}$$

Multiply -5 and 2 to get -10.

$$\frac{-10+4\sqrt{2}+5\left(-1\right)\sqrt{2}+7}{7}$$

Multiply 5 and -1 to get -5.

$$\frac{-10+4\sqrt{2}-5\sqrt{2}+7}{7}$$

Combine 4\sqrt{2} and -5\sqrt{2} to get -\sqrt{2}.

$$\frac{-10-\sqrt{2}+7}{7}$$

Add -10 and 7 to get -3.

$$\frac{-3-\sqrt{2}}{7}$$