Solve frac{3-2sqrt{2}}{5-sqrt{2}}

$$\frac { 3 - 2 \sqrt { 2 } } { 5 - \sqrt { 2 } }$$

Evaluate

$$\frac{11-7\sqrt{2}}{23}\approx 0.047848046$$

Solution Steps

Rationalize the denominator of \frac{3-2\sqrt{2}}{5-\sqrt{2}} by multiplying numerator and denominator by 5+\sqrt{2}.

$$\frac{\left(3-2\sqrt{2}\right)\left(5+\sqrt{2}\right)}{\left(5-\sqrt{2}\right)\left(5+\sqrt{2}\right)}$$

Consider \left(5-\sqrt{2}\right)\left(5+\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}.

$$\frac{\left(3-2\sqrt{2}\right)\left(5+\sqrt{2}\right)}{5^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 5. Square \sqrt{2}.

$$\frac{\left(3-2\sqrt{2}\right)\left(5+\sqrt{2}\right)}{25-2}$$

Subtract 2 from 25 to get 23.

$$\frac{\left(3-2\sqrt{2}\right)\left(5+\sqrt{2}\right)}{23}$$

Apply the distributive property by multiplying each term of 3-2\sqrt{2} by each term of 5+\sqrt{2}.

$$\frac{15+3\sqrt{2}-10\sqrt{2}-2\left(\sqrt{2}\right)^{2}}{23}$$

Combine 3\sqrt{2} and -10\sqrt{2} to get -7\sqrt{2}.

$$\frac{15-7\sqrt{2}-2\left(\sqrt{2}\right)^{2}}{23}$$

The square of \sqrt{2} is 2.

$$\frac{15-7\sqrt{2}-2\times 2}{23}$$

Multiply -2 and 2 to get -4.

$$\frac{15-7\sqrt{2}-4}{23}$$

Subtract 4 from 15 to get 11.

$$\frac{11-7\sqrt{2}}{23}$$