Solve frac{sqrt{5}-1}{sqrt{2}+1}

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

Evaluate

$$\sqrt{10}+1-\sqrt{2}-\sqrt{5}\approx 0.51199612$$

Solution Steps

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

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

Consider \left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{2}. Square 1.

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

Subtract 1 from 2 to get 1.

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

Anything divided by one gives itself.

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

Apply the distributive property by multiplying each term of \sqrt{5}-1 by each term of \sqrt{2}-1.

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

To multiply \sqrt{5} and \sqrt{2}, multiply the numbers under the square root.

$$\sqrt{10}-\sqrt{5}-\sqrt{2}+1$$