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

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

Evaluate

$$-\frac{\sqrt{2}\left(\sqrt{5}-3\right)}{4}\approx 0.270090757$$

Solution Steps

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

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

Consider \left(\sqrt{5}+3\right)\left(\sqrt{5}-3\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Square \sqrt{5}. Square 3.

$$\frac{\sqrt{2}\left(\sqrt{5}-3\right)}{5-9}$$

Subtract 9 from 5 to get -4.

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

Use the distributive property to multiply \sqrt{2} by \sqrt{5}-3.

$$\frac{\sqrt{2}\sqrt{5}-3\sqrt{2}}{-4}$$

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

$$\frac{\sqrt{10}-3\sqrt{2}}{-4}$$

Multiply both numerator and denominator by -1.

$$\frac{-\sqrt{10}+3\sqrt{2}}{4}$$