Solve frac{sqrt{8}}{sqrt{5}+sqrt{3}}

$$\frac{ \sqrt{ 8 } }{ \sqrt{ 5 } + \sqrt{ 3 } }$$

Evaluate

$$\sqrt{10}-\sqrt{6}\approx 0.712787917$$

Solution Steps

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

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

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

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

Consider \left(\sqrt{5}+\sqrt{3}\right)\left(\sqrt{5}-\sqrt{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}.

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

Square \sqrt{5}. Square \sqrt{3}.

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

Subtract 3 from 5 to get 2.

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

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

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

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

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

To multiply \sqrt{3} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{2\sqrt{10}-2\sqrt{6}}{2}$$

Divide each term of 2\sqrt{10}-2\sqrt{6} by 2 to get \sqrt{10}-\sqrt{6}.

$$\sqrt{10}-\sqrt{6}$$