Solve 2sqrt{5}+frac{4/sqrt{6}}{sqrt{2/15}}-sqrt{45}

$$2 \sqrt{ 5 } + \frac{ \frac{ 4 }{ \sqrt{ 6 } } }{ \sqrt{ \frac{ 2 }{ 15 } } } - \sqrt{ 45 }$$

Evaluate

$$\sqrt{5}\approx 2.236067977$$

Solution Steps

Rationalize the denominator of \frac{4}{\sqrt{6}} by multiplying numerator and denominator by \sqrt{6}.

$$2\sqrt{5}+\frac{\frac{4\sqrt{6}}{\left(\sqrt{6}\right)^{2}}}{\sqrt{\frac{2}{15}}}-\sqrt{45}$$

The square of \sqrt{6} is 6.

$$2\sqrt{5}+\frac{\frac{4\sqrt{6}}{6}}{\sqrt{\frac{2}{15}}}-\sqrt{45}$$

Divide 4\sqrt{6} by 6 to get \frac{2}{3}\sqrt{6}.

$$2\sqrt{5}+\frac{\frac{2}{3}\sqrt{6}}{\sqrt{\frac{2}{15}}}-\sqrt{45}$$

Rewrite the square root of the division \sqrt{\frac{2}{15}} as the division of square roots \frac{\sqrt{2}}{\sqrt{15}}.

$$2\sqrt{5}+\frac{\frac{2}{3}\sqrt{6}}{\frac{\sqrt{2}}{\sqrt{15}}}-\sqrt{45}$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{15}} by multiplying numerator and denominator by \sqrt{15}.

$$2\sqrt{5}+\frac{\frac{2}{3}\sqrt{6}}{\frac{\sqrt{2}\sqrt{15}}{\left(\sqrt{15}\right)^{2}}}-\sqrt{45}$$

The square of \sqrt{15} is 15.

$$2\sqrt{5}+\frac{\frac{2}{3}\sqrt{6}}{\frac{\sqrt{2}\sqrt{15}}{15}}-\sqrt{45}$$

To multiply \sqrt{2} and \sqrt{15}, multiply the numbers under the square root.

$$2\sqrt{5}+\frac{\frac{2}{3}\sqrt{6}}{\frac{\sqrt{30}}{15}}-\sqrt{45}$$

Divide \frac{2}{3}\sqrt{6} by \frac{\sqrt{30}}{15} by multiplying \frac{2}{3}\sqrt{6} by the reciprocal of \frac{\sqrt{30}}{15}.

$$2\sqrt{5}+\frac{\frac{2}{3}\sqrt{6}\times 15}{\sqrt{30}}-\sqrt{45}$$

Rationalize the denominator of \frac{\frac{2}{3}\sqrt{6}\times 15}{\sqrt{30}} by multiplying numerator and denominator by \sqrt{30}.

$$2\sqrt{5}+\frac{\frac{2}{3}\sqrt{6}\times 15\sqrt{30}}{\left(\sqrt{30}\right)^{2}}-\sqrt{45}$$

The square of \sqrt{30} is 30.

$$2\sqrt{5}+\frac{\frac{2}{3}\sqrt{6}\times 15\sqrt{30}}{30}-\sqrt{45}$$

Express \frac{2}{3}\times 15 as a single fraction.

$$2\sqrt{5}+\frac{\frac{2\times 15}{3}\sqrt{6}\sqrt{30}}{30}-\sqrt{45}$$

Multiply 2 and 15 to get 30.

$$2\sqrt{5}+\frac{\frac{30}{3}\sqrt{6}\sqrt{30}}{30}-\sqrt{45}$$

Divide 30 by 3 to get 10.

$$2\sqrt{5}+\frac{10\sqrt{6}\sqrt{30}}{30}-\sqrt{45}$$

Factor 30=6\times 5. Rewrite the square root of the product \sqrt{6\times 5} as the product of square roots \sqrt{6}\sqrt{5}.

$$2\sqrt{5}+\frac{10\sqrt{6}\sqrt{6}\sqrt{5}}{30}-\sqrt{45}$$

Multiply \sqrt{6} and \sqrt{6} to get 6.

$$2\sqrt{5}+\frac{10\times 6\sqrt{5}}{30}-\sqrt{45}$$

Multiply 10 and 6 to get 60.

$$2\sqrt{5}+\frac{60\sqrt{5}}{30}-\sqrt{45}$$

Divide 60\sqrt{5} by 30 to get 2\sqrt{5}.

$$2\sqrt{5}+2\sqrt{5}-\sqrt{45}$$

Combine 2\sqrt{5} and 2\sqrt{5} to get 4\sqrt{5}.

$$4\sqrt{5}-\sqrt{45}$$

Factor 45=3^{2}\times 5. Rewrite the square root of the product \sqrt{3^{2}\times 5} as the product of square roots \sqrt{3^{2}}\sqrt{5}. Take the square root of 3^{2}.

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

Combine 4\sqrt{5} and -3\sqrt{5} to get \sqrt{5}.

$$\sqrt{5}$$