Solve 2(2+1)/(2-sqrt{2)}=

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

Evaluate

$$3\sqrt{2}+6\approx 10.242640687$$

Solution Steps

Add 2 and 1 to get 3.

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

Multiply 2 and 3 to get 6.

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

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

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

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

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

$$\frac{6\left(2+\sqrt{2}\right)}{2}$$

Divide 6\left(2+\sqrt{2}\right) by 2 to get 3\left(2+\sqrt{2}\right).

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

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

$$6+3\sqrt{2}$$