Solve 1/3-2sqrt{2}+3-2sqrt{2}

$$\frac{ 1 }{ 3-2 \sqrt{ 2 } } +3-2 \sqrt{ 2 }$$

Evaluate

$$6$$

Solution Steps

Rationalize the denominator of \frac{1}{3-2\sqrt{2}} by multiplying numerator and denominator by 3+2\sqrt{2}.

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

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}.

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

Calculate 3 to the power of 2 and get 9.

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

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

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

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

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

The square of \sqrt{2} is 2.

$$\frac{3+2\sqrt{2}}{9-4\times 2}+3-2\sqrt{2}$$

Multiply 4 and 2 to get 8.

$$\frac{3+2\sqrt{2}}{9-8}+3-2\sqrt{2}$$

Subtract 8 from 9 to get 1.

$$\frac{3+2\sqrt{2}}{1}+3-2\sqrt{2}$$

Anything divided by one gives itself.

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

Add 3 and 3 to get 6.

$$6+2\sqrt{2}-2\sqrt{2}$$

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

$$6$$

Factor

$$2\times 3$$