Solve 5/1+sqrt{2}

$$\frac { 5 } { 1 + \sqrt { 2 } }$$

Evaluate

$$5\left(\sqrt{2}-1\right)\approx 2.071067812$$

Solution Steps

Rationalize the denominator of \frac{5}{1+\sqrt{2}} by multiplying numerator and denominator by 1-\sqrt{2}.

$$\frac{5\left(1-\sqrt{2}\right)}{\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)}$$

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}.

$$\frac{5\left(1-\sqrt{2}\right)}{1^{2}-\left(\sqrt{2}\right)^{2}}$$

Square 1. Square \sqrt{2}.

$$\frac{5\left(1-\sqrt{2}\right)}{1-2}$$

Subtract 2 from 1 to get -1.

$$\frac{5\left(1-\sqrt{2}\right)}{-1}$$

Anything divided by -1 gives its opposite.

$$-5\left(1-\sqrt{2}\right)$$

Use the distributive property to multiply -5 by 1-\sqrt{2}.

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