Solve frac{sqrt{2}}{sqrt{2}+1}-1/sqrt{2-1}

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

Evaluate

$$1-2\sqrt{2}\approx -1.828427125$$

Solution Steps

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

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

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}.

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

Square \sqrt{2}. Square 1.

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

Subtract 1 from 2 to get 1.

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

Anything divided by one gives itself.

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

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

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

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}.

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

Square \sqrt{2}. Square 1.

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

Subtract 1 from 2 to get 1.

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

Anything divided by one gives itself.

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

To find the opposite of \sqrt{2}+1, find the opposite of each term.

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

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

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

The square of \sqrt{2} is 2.

$$2-\sqrt{2}-\sqrt{2}-1$$

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

$$2-2\sqrt{2}-1$$

Subtract 1 from 2 to get 1.

$$1-2\sqrt{2}$$