Solve frac{3sqrt{45}}{sqrt{1/5}}2/3sqrt{22/3}

$$\frac{ 3 \sqrt{ 45 } }{ \sqrt{ \frac{ 1 }{ 5 } } } \frac{ 2 }{ 3 } \sqrt{ 2 \frac{ 2 }{ 3 } }$$

Evaluate

$$20\sqrt{6}\approx 48.989794856$$

Solution Steps

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}.

$$\frac{3\times 3\sqrt{5}}{\sqrt{\frac{1}{5}}}\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

Multiply 3 and 3 to get 9.

$$\frac{9\sqrt{5}}{\sqrt{\frac{1}{5}}}\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

Rewrite the square root of the division \sqrt{\frac{1}{5}} as the division of square roots \frac{\sqrt{1}}{\sqrt{5}}.

$$\frac{9\sqrt{5}}{\frac{\sqrt{1}}{\sqrt{5}}}\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

Calculate the square root of 1 and get 1.

$$\frac{9\sqrt{5}}{\frac{1}{\sqrt{5}}}\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

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

$$\frac{9\sqrt{5}}{\frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

The square of \sqrt{5} is 5.

$$\frac{9\sqrt{5}}{\frac{\sqrt{5}}{5}}\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

Divide 9\sqrt{5} by \frac{\sqrt{5}}{5} by multiplying 9\sqrt{5} by the reciprocal of \frac{\sqrt{5}}{5}.

$$\frac{9\sqrt{5}\times 5}{\sqrt{5}}\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

Rationalize the denominator of \frac{9\sqrt{5}\times 5}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{9\sqrt{5}\times 5\sqrt{5}}{\left(\sqrt{5}\right)^{2}}\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

The square of \sqrt{5} is 5.

$$\frac{9\sqrt{5}\times 5\sqrt{5}}{5}\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

Multiply 9 and 5 to get 45.

$$\frac{45\sqrt{5}\sqrt{5}}{5}\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

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

$$\frac{45\times 5}{5}\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

Multiply 45 and 5 to get 225.

$$\frac{225}{5}\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

Divide 225 by 5 to get 45.

$$45\times \frac{2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

Express 45\times \frac{2}{3} as a single fraction.

$$\frac{45\times 2}{3}\sqrt{\frac{2\times 3+2}{3}}$$

Multiply 45 and 2 to get 90.

$$\frac{90}{3}\sqrt{\frac{2\times 3+2}{3}}$$

Divide 90 by 3 to get 30.

$$30\sqrt{\frac{2\times 3+2}{3}}$$

Multiply 2 and 3 to get 6.

$$30\sqrt{\frac{6+2}{3}}$$

Add 6 and 2 to get 8.

$$30\sqrt{\frac{8}{3}}$$

Rewrite the square root of the division \sqrt{\frac{8}{3}} as the division of square roots \frac{\sqrt{8}}{\sqrt{3}}.

$$30\times \frac{\sqrt{8}}{\sqrt{3}}$$

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}.

$$30\times \frac{2\sqrt{2}}{\sqrt{3}}$$

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

$$30\times \frac{2\sqrt{2}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$30\times \frac{2\sqrt{2}\sqrt{3}}{3}$$

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

$$30\times \frac{2\sqrt{6}}{3}$$

Cancel out 3, the greatest common factor in 30 and 3.

$$10\times 2\sqrt{6}$$

Multiply 10 and 2 to get 20.

$$20\sqrt{6}$$