Solve frac{2+5sqrt{2}}{2-sqrt{2}}

$$\frac { 2 + 5 \sqrt { 2 } } { 2 - \sqrt { 2 } }$$

Evaluate

$$6\sqrt{2}+7\approx 15.485281374$$

Solution Steps

Rationalize the denominator of \frac{2+5\sqrt{2}}{2-\sqrt{2}} by multiplying numerator and denominator by 2+\sqrt{2}.

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

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}.

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

Square 2. Square \sqrt{2}.

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

Subtract 2 from 4 to get 2.

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

Apply the distributive property by multiplying each term of 2+5\sqrt{2} by each term of 2+\sqrt{2}.

$$\frac{4+2\sqrt{2}+10\sqrt{2}+5\left(\sqrt{2}\right)^{2}}{2}$$

Combine 2\sqrt{2} and 10\sqrt{2} to get 12\sqrt{2}.

$$\frac{4+12\sqrt{2}+5\left(\sqrt{2}\right)^{2}}{2}$$

The square of \sqrt{2} is 2.

$$\frac{4+12\sqrt{2}+5\times 2}{2}$$

Multiply 5 and 2 to get 10.

$$\frac{4+12\sqrt{2}+10}{2}$$

Add 4 and 10 to get 14.

$$\frac{14+12\sqrt{2}}{2}$$

Divide each term of 14+12\sqrt{2} by 2 to get 7+6\sqrt{2}.

$$7+6\sqrt{2}$$

Factor

$$6 \sqrt{2} + 7 = 15.485281374$$