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

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

Evaluate

$$4\sqrt{2}-5\approx 0.656854249$$

Solution Steps

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

$$\frac{\left(3-\sqrt{2}\right)\left(1-\sqrt{2}\right)}{\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)}$$

Consider \left(1+\sqrt{2}\right)\left(1-\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}.

$$\frac{\left(3-\sqrt{2}\right)\left(1-\sqrt{2}\right)}{1^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 1. Square \sqrt{2}.

$$\frac{\left(3-\sqrt{2}\right)\left(1-\sqrt{2}\right)}{1-2}$$

Subtract 2 from 1 to get -1.

$$\frac{\left(3-\sqrt{2}\right)\left(1-\sqrt{2}\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-\left(3-\sqrt{2}\right)\left(1-\sqrt{2}\right)$$

Apply the distributive property by multiplying each term of 3-\sqrt{2} by each term of 1-\sqrt{2}.

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

Combine -3\sqrt{2} and -\sqrt{2} to get -4\sqrt{2}.

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

The square of \sqrt{2} is 2.

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

Add 3 and 2 to get 5.

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

To find the opposite of 5-4\sqrt{2}, find the opposite of each term.

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

The opposite of -4\sqrt{2} is 4\sqrt{2}.

$$-5+4\sqrt{2}$$