Solve frac{3+sqrt{3}}{sqrt{2}}*(frac{1+sqrt{3}}{sqrt{5}})

$$\frac { 3 + \sqrt { 3 } } { \sqrt { 2 } } \cdot ( \frac { 1 + \sqrt { 3 } } { \sqrt { 5 } } )$$

Evaluate

$$\frac{2\sqrt{30}+3\sqrt{10}}{5}\approx 4.088256826$$

Solution Steps

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

$$\frac{\left(3+\sqrt{3}\right)\sqrt{2}}{\left(\sqrt{2}\right)^{2}}\times \frac{1+\sqrt{3}}{\sqrt{5}}$$

The square of \sqrt{2} is 2.

$$\frac{\left(3+\sqrt{3}\right)\sqrt{2}}{2}\times \frac{1+\sqrt{3}}{\sqrt{5}}$$

Rationalize the denominator of \frac{1+\sqrt{3}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\left(3+\sqrt{3}\right)\sqrt{2}}{2}\times \frac{\left(1+\sqrt{3}\right)\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{\left(3+\sqrt{3}\right)\sqrt{2}}{2}\times \frac{\left(1+\sqrt{3}\right)\sqrt{5}}{5}$$

Multiply \frac{\left(3+\sqrt{3}\right)\sqrt{2}}{2} times \frac{\left(1+\sqrt{3}\right)\sqrt{5}}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(3+\sqrt{3}\right)\sqrt{2}\left(1+\sqrt{3}\right)\sqrt{5}}{2\times 5}$$

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

$$\frac{\left(3+\sqrt{3}\right)\sqrt{10}\left(1+\sqrt{3}\right)}{2\times 5}$$

Multiply 2 and 5 to get 10.

$$\frac{\left(3+\sqrt{3}\right)\sqrt{10}\left(1+\sqrt{3}\right)}{10}$$

Use the distributive property to multiply 3+\sqrt{3} by \sqrt{10}.

$$\frac{\left(3\sqrt{10}+\sqrt{3}\sqrt{10}\right)\left(1+\sqrt{3}\right)}{10}$$

To multiply \sqrt{3} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{\left(3\sqrt{10}+\sqrt{30}\right)\left(1+\sqrt{3}\right)}{10}$$

Apply the distributive property by multiplying each term of 3\sqrt{10}+\sqrt{30} by each term of 1+\sqrt{3}.

$$\frac{3\sqrt{10}+3\sqrt{10}\sqrt{3}+\sqrt{30}+\sqrt{30}\sqrt{3}}{10}$$

To multiply \sqrt{10} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{3\sqrt{10}+3\sqrt{30}+\sqrt{30}+\sqrt{30}\sqrt{3}}{10}$$

Combine 3\sqrt{30} and \sqrt{30} to get 4\sqrt{30}.

$$\frac{3\sqrt{10}+4\sqrt{30}+\sqrt{30}\sqrt{3}}{10}$$

Factor 30=3\times 10. Rewrite the square root of the product \sqrt{3\times 10} as the product of square roots \sqrt{3}\sqrt{10}.

$$\frac{3\sqrt{10}+4\sqrt{30}+\sqrt{3}\sqrt{10}\sqrt{3}}{10}$$

Multiply \sqrt{3} and \sqrt{3} to get 3.

$$\frac{3\sqrt{10}+4\sqrt{30}+3\sqrt{10}}{10}$$

Combine 3\sqrt{10} and 3\sqrt{10} to get 6\sqrt{10}.

$$\frac{6\sqrt{10}+4\sqrt{30}}{10}$$

Factor

$$\frac{2 \sqrt{30} + 3 \sqrt{10}}{5} = 4.088256826121691$$