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

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

Evaluate

$$\frac{1}{2}-\sqrt{2}\approx -0.914213562$$

Solution Steps

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

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

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

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

Calculate 2 to the power of 2 and get 4.

$$\frac{\left(5-3\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{4-\left(-2\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(5-3\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{4-\left(-2\right)^{2}\left(\sqrt{2}\right)^{2}}$$

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

$$\frac{\left(5-3\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{4-4\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(5-3\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{4-4\times 2}$$

Multiply 4 and 2 to get 8.

$$\frac{\left(5-3\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{4-8}$$

Subtract 8 from 4 to get -4.

$$\frac{\left(5-3\sqrt{2}\right)\left(2+2\sqrt{2}\right)}{-4}$$

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

$$\frac{10+10\sqrt{2}-6\sqrt{2}-6\left(\sqrt{2}\right)^{2}}{-4}$$

Combine 10\sqrt{2} and -6\sqrt{2} to get 4\sqrt{2}.

$$\frac{10+4\sqrt{2}-6\left(\sqrt{2}\right)^{2}}{-4}$$

The square of \sqrt{2} is 2.

$$\frac{10+4\sqrt{2}-6\times 2}{-4}$$

Multiply -6 and 2 to get -12.

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

Subtract 12 from 10 to get -2.

$$\frac{-2+4\sqrt{2}}{-4}$$