Solve frac{44+4sqrt{5}}{(sqrt{10}+sqrt{2})}

$$\frac { 44 + 4 \sqrt { 5 } } { ( \sqrt { 10 } + \sqrt { 2 } ) }$$

Evaluate

$$5\sqrt{10}-3\sqrt{2}\approx 11.568747614$$

Solution Steps

Rationalize the denominator of \frac{44+4\sqrt{5}}{\sqrt{10}+\sqrt{2}} by multiplying numerator and denominator by \sqrt{10}-\sqrt{2}.

$$\frac{\left(44+4\sqrt{5}\right)\left(\sqrt{10}-\sqrt{2}\right)}{\left(\sqrt{10}+\sqrt{2}\right)\left(\sqrt{10}-\sqrt{2}\right)}$$

Consider \left(\sqrt{10}+\sqrt{2}\right)\left(\sqrt{10}-\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}.

$$\frac{\left(44+4\sqrt{5}\right)\left(\sqrt{10}-\sqrt{2}\right)}{\left(\sqrt{10}\right)^{2}-\left(\sqrt{2}\right)^{2}}$$

Square \sqrt{10}. Square \sqrt{2}.

$$\frac{\left(44+4\sqrt{5}\right)\left(\sqrt{10}-\sqrt{2}\right)}{10-2}$$

Subtract 2 from 10 to get 8.

$$\frac{\left(44+4\sqrt{5}\right)\left(\sqrt{10}-\sqrt{2}\right)}{8}$$

Apply the distributive property by multiplying each term of 44+4\sqrt{5} by each term of \sqrt{10}-\sqrt{2}.

$$\frac{44\sqrt{10}-44\sqrt{2}+4\sqrt{5}\sqrt{10}-4\sqrt{5}\sqrt{2}}{8}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$\frac{44\sqrt{10}-44\sqrt{2}+4\sqrt{5}\sqrt{5}\sqrt{2}-4\sqrt{5}\sqrt{2}}{8}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{44\sqrt{10}-44\sqrt{2}+4\times 5\sqrt{2}-4\sqrt{5}\sqrt{2}}{8}$$

Multiply 4 and 5 to get 20.

$$\frac{44\sqrt{10}-44\sqrt{2}+20\sqrt{2}-4\sqrt{5}\sqrt{2}}{8}$$

Combine -44\sqrt{2} and 20\sqrt{2} to get -24\sqrt{2}.

$$\frac{44\sqrt{10}-24\sqrt{2}-4\sqrt{5}\sqrt{2}}{8}$$

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

$$\frac{44\sqrt{10}-24\sqrt{2}-4\sqrt{10}}{8}$$

Combine 44\sqrt{10} and -4\sqrt{10} to get 40\sqrt{10}.

$$\frac{40\sqrt{10}-24\sqrt{2}}{8}$$

Divide each term of 40\sqrt{10}-24\sqrt{2} by 8 to get 5\sqrt{10}-3\sqrt{2}.

$$5\sqrt{10}-3\sqrt{2}$$

Factor

$$5 \sqrt{10} - 3 \sqrt{2} = 11.568747614$$