Solve 6/sqrt{2}-frac{sqrt{2}}{3-2sqrt{2}}

$$\frac { 6 } { \sqrt { 2 } } - \frac { \sqrt { 2 } } { 3 - 2 \sqrt { 2 } }$$

Evaluate

$$-4$$

Solution Steps

Rationalize the denominator of \frac{6}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

$$\frac{6\sqrt{2}}{2}-\frac{\sqrt{2}}{3-2\sqrt{2}}$$

Divide 6\sqrt{2} by 2 to get 3\sqrt{2}.

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

Rationalize the denominator of \frac{\sqrt{2}}{3-2\sqrt{2}} by multiplying numerator and denominator by 3+2\sqrt{2}.

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

Consider \left(3-2\sqrt{2}\right)\left(3+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}.

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

Calculate 3 to the power of 2 and get 9.

$$3\sqrt{2}-\frac{\sqrt{2}\left(3+2\sqrt{2}\right)}{9-\left(-2\sqrt{2}\right)^{2}}$$

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

$$3\sqrt{2}-\frac{\sqrt{2}\left(3+2\sqrt{2}\right)}{9-\left(-2\right)^{2}\left(\sqrt{2}\right)^{2}}$$

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

$$3\sqrt{2}-\frac{\sqrt{2}\left(3+2\sqrt{2}\right)}{9-4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$3\sqrt{2}-\frac{\sqrt{2}\left(3+2\sqrt{2}\right)}{9-4\times 2}$$

Multiply 4 and 2 to get 8.

$$3\sqrt{2}-\frac{\sqrt{2}\left(3+2\sqrt{2}\right)}{9-8}$$

Subtract 8 from 9 to get 1.

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

Anything divided by one gives itself.

$$3\sqrt{2}-\sqrt{2}\left(3+2\sqrt{2}\right)$$

Use the distributive property to multiply \sqrt{2} by 3+2\sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Multiply 2 and 2 to get 4.

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

To find the opposite of 3\sqrt{2}+4, find the opposite of each term.

$$3\sqrt{2}-3\sqrt{2}-4$$

Combine 3\sqrt{2} and -3\sqrt{2} to get 0.

$$-4$$

Factor

$$-4$$