Solve sqrt{frac{2sqrt{273}+6}{3}(frac{2sqrt{273}+6}{3}-frac{2sqrt{273}}{3})(frac{2sqrt{273}+6}{3}-frac{2sqrt{273}}{3})(frac{2sqrt{273}+6}{3}-4)}

$$\sqrt{ \frac{ 2 \sqrt{ 273 } +6 }{ 3 } ( \frac{ 2 \sqrt{ 273 } +6 }{ 3 } - \frac{ 2 \sqrt{ 273 } }{ 3 } )( \frac{ 2 \sqrt{ 273 } +6 }{ 3 } - \frac{ 2 \sqrt{ 273 } }{ 3 } )( \frac{ 2 \sqrt{ 273 } +6 }{ 3 } -4) }$$

Evaluate

$$\frac{8\sqrt{66}}{3}\approx 21.664102412$$

Short Solution Steps

Multiply \frac{2\sqrt{273}+6}{3}-\frac{2\sqrt{273}}{3} and \frac{2\sqrt{273}+6}{3}-\frac{2\sqrt{273}}{3} to get \left(\frac{2\sqrt{273}+6}{3}-\frac{2\sqrt{273}}{3}\right)^{2}.

$$\sqrt{\frac{2\sqrt{273}+6}{3}\left(\frac{2\sqrt{273}+6}{3}-\frac{2\sqrt{273}}{3}\right)^{2}\left(\frac{2\sqrt{273}+6}{3}-4\right)}$$

Since \frac{2\sqrt{273}+6}{3} and \frac{2\sqrt{273}}{3} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{2\sqrt{273}+6}{3}\times \left(\frac{2\sqrt{273}+6-2\sqrt{273}}{3}\right)^{2}\left(\frac{2\sqrt{273}+6}{3}-4\right)}$$

Do the calculations in 2\sqrt{273}+6-2\sqrt{273}.

$$\sqrt{\frac{2\sqrt{273}+6}{3}\times \left(\frac{6}{3}\right)^{2}\left(\frac{2\sqrt{273}+6}{3}-4\right)}$$

To raise \frac{6}{3} to a power, raise both numerator and denominator to the power and then divide.

$$\sqrt{\frac{2\sqrt{273}+6}{3}\times \frac{6^{2}}{3^{2}}\left(\frac{2\sqrt{273}+6}{3}-4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 4 times \frac{3}{3}.

$$\sqrt{\frac{2\sqrt{273}+6}{3}\times \frac{6^{2}}{3^{2}}\left(\frac{2\sqrt{273}+6}{3}-\frac{4\times 3}{3}\right)}$$

Since \frac{2\sqrt{273}+6}{3} and \frac{4\times 3}{3} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{2\sqrt{273}+6}{3}\times \frac{6^{2}}{3^{2}}\times \frac{2\sqrt{273}+6-4\times 3}{3}}$$

Do the multiplications in 2\sqrt{273}+6-4\times 3.

$$\sqrt{\frac{2\sqrt{273}+6}{3}\times \frac{6^{2}}{3^{2}}\times \frac{2\sqrt{273}+6-12}{3}}$$

Do the calculations in 2\sqrt{273}+6-12.

$$\sqrt{\frac{2\sqrt{273}+6}{3}\times \frac{6^{2}}{3^{2}}\times \frac{2\sqrt{273}-6}{3}}$$

Multiply \frac{2\sqrt{273}+6}{3} times \frac{6^{2}}{3^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\left(2\sqrt{273}+6\right)\times 6^{2}}{3\times 3^{2}}\times \frac{2\sqrt{273}-6}{3}}$$

Multiply \frac{\left(2\sqrt{273}+6\right)\times 6^{2}}{3\times 3^{2}} times \frac{2\sqrt{273}-6}{3} by multiplying numerator times numerator and denominator times denominator.

$$\sqrt{\frac{\left(2\sqrt{273}+6\right)\times 6^{2}\left(2\sqrt{273}-6\right)}{3\times 3^{2}\times 3}}$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$\sqrt{\frac{\left(2\sqrt{273}+6\right)\times 6^{2}\left(2\sqrt{273}-6\right)}{3^{3}\times 3}}$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$\sqrt{\frac{\left(2\sqrt{273}+6\right)\times 6^{2}\left(2\sqrt{273}-6\right)}{3^{4}}}$$

Calculate 6 to the power of 2 and get 36.

$$\sqrt{\frac{\left(2\sqrt{273}+6\right)\times 36\left(2\sqrt{273}-6\right)}{3^{4}}}$$

Calculate 3 to the power of 4 and get 81.

$$\sqrt{\frac{\left(2\sqrt{273}+6\right)\times 36\left(2\sqrt{273}-6\right)}{81}}$$

Divide \left(2\sqrt{273}+6\right)\times 36\left(2\sqrt{273}-6\right) by 81 to get \left(2\sqrt{273}+6\right)\times \frac{4}{9}\left(2\sqrt{273}-6\right).

$$\sqrt{\left(2\sqrt{273}+6\right)\times \frac{4}{9}\left(2\sqrt{273}-6\right)}$$

Use the distributive property to multiply 2\sqrt{273}+6 by \frac{4}{9}.

$$\sqrt{\left(2\sqrt{273}\times \frac{4}{9}+6\times \frac{4}{9}\right)\left(2\sqrt{273}-6\right)}$$

Express 2\times \frac{4}{9} as a single fraction.

$$\sqrt{\left(\frac{2\times 4}{9}\sqrt{273}+6\times \frac{4}{9}\right)\left(2\sqrt{273}-6\right)}$$

Multiply 2 and 4 to get 8.

$$\sqrt{\left(\frac{8}{9}\sqrt{273}+6\times \frac{4}{9}\right)\left(2\sqrt{273}-6\right)}$$

Express 6\times \frac{4}{9} as a single fraction.

$$\sqrt{\left(\frac{8}{9}\sqrt{273}+\frac{6\times 4}{9}\right)\left(2\sqrt{273}-6\right)}$$

Multiply 6 and 4 to get 24.

$$\sqrt{\left(\frac{8}{9}\sqrt{273}+\frac{24}{9}\right)\left(2\sqrt{273}-6\right)}$$

Reduce the fraction \frac{24}{9} to lowest terms by extracting and canceling out 3.

$$\sqrt{\left(\frac{8}{9}\sqrt{273}+\frac{8}{3}\right)\left(2\sqrt{273}-6\right)}$$

Apply the distributive property by multiplying each term of \frac{8}{9}\sqrt{273}+\frac{8}{3} by each term of 2\sqrt{273}-6.

$$\sqrt{\frac{8}{9}\sqrt{273}\times 2\sqrt{273}+\frac{8}{9}\sqrt{273}\left(-6\right)+\frac{8}{3}\times 2\sqrt{273}+\frac{8}{3}\left(-6\right)}$$

Multiply \sqrt{273} and \sqrt{273} to get 273.

$$\sqrt{\frac{8}{9}\times 273\times 2+\frac{8}{9}\sqrt{273}\left(-6\right)+\frac{8}{3}\times 2\sqrt{273}+\frac{8}{3}\left(-6\right)}$$

Express \frac{8}{9}\times 273 as a single fraction.

$$\sqrt{\frac{8\times 273}{9}\times 2+\frac{8}{9}\sqrt{273}\left(-6\right)+\frac{8}{3}\times 2\sqrt{273}+\frac{8}{3}\left(-6\right)}$$

Multiply 8 and 273 to get 2184.

$$\sqrt{\frac{2184}{9}\times 2+\frac{8}{9}\sqrt{273}\left(-6\right)+\frac{8}{3}\times 2\sqrt{273}+\frac{8}{3}\left(-6\right)}$$

Reduce the fraction \frac{2184}{9} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{728}{3}\times 2+\frac{8}{9}\sqrt{273}\left(-6\right)+\frac{8}{3}\times 2\sqrt{273}+\frac{8}{3}\left(-6\right)}$$

Express \frac{728}{3}\times 2 as a single fraction.

$$\sqrt{\frac{728\times 2}{3}+\frac{8}{9}\sqrt{273}\left(-6\right)+\frac{8}{3}\times 2\sqrt{273}+\frac{8}{3}\left(-6\right)}$$

Multiply 728 and 2 to get 1456.

$$\sqrt{\frac{1456}{3}+\frac{8}{9}\sqrt{273}\left(-6\right)+\frac{8}{3}\times 2\sqrt{273}+\frac{8}{3}\left(-6\right)}$$

Express \frac{8}{9}\left(-6\right) as a single fraction.

$$\sqrt{\frac{1456}{3}+\frac{8\left(-6\right)}{9}\sqrt{273}+\frac{8}{3}\times 2\sqrt{273}+\frac{8}{3}\left(-6\right)}$$

Multiply 8 and -6 to get -48.

$$\sqrt{\frac{1456}{3}+\frac{-48}{9}\sqrt{273}+\frac{8}{3}\times 2\sqrt{273}+\frac{8}{3}\left(-6\right)}$$

Reduce the fraction \frac{-48}{9} to lowest terms by extracting and canceling out 3.

$$\sqrt{\frac{1456}{3}-\frac{16}{3}\sqrt{273}+\frac{8}{3}\times 2\sqrt{273}+\frac{8}{3}\left(-6\right)}$$

Express \frac{8}{3}\times 2 as a single fraction.

$$\sqrt{\frac{1456}{3}-\frac{16}{3}\sqrt{273}+\frac{8\times 2}{3}\sqrt{273}+\frac{8}{3}\left(-6\right)}$$

Multiply 8 and 2 to get 16.

$$\sqrt{\frac{1456}{3}-\frac{16}{3}\sqrt{273}+\frac{16}{3}\sqrt{273}+\frac{8}{3}\left(-6\right)}$$

Combine -\frac{16}{3}\sqrt{273} and \frac{16}{3}\sqrt{273} to get 0.

$$\sqrt{\frac{1456}{3}+\frac{8}{3}\left(-6\right)}$$

Express \frac{8}{3}\left(-6\right) as a single fraction.

$$\sqrt{\frac{1456}{3}+\frac{8\left(-6\right)}{3}}$$

Multiply 8 and -6 to get -48.

$$\sqrt{\frac{1456}{3}+\frac{-48}{3}}$$

Divide -48 by 3 to get -16.

$$\sqrt{\frac{1456}{3}-16}$$

Convert 16 to fraction \frac{48}{3}.

$$\sqrt{\frac{1456}{3}-\frac{48}{3}}$$

Since \frac{1456}{3} and \frac{48}{3} have the same denominator, subtract them by subtracting their numerators.

$$\sqrt{\frac{1456-48}{3}}$$

Subtract 48 from 1456 to get 1408.

$$\sqrt{\frac{1408}{3}}$$

Rewrite the square root of the division \sqrt{\frac{1408}{3}} as the division of square roots \frac{\sqrt{1408}}{\sqrt{3}}.

$$\frac{\sqrt{1408}}{\sqrt{3}}$$

Factor 1408=8^{2}\times 22. Rewrite the square root of the product \sqrt{8^{2}\times 22} as the product of square roots \sqrt{8^{2}}\sqrt{22}. Take the square root of 8^{2}.

$$\frac{8\sqrt{22}}{\sqrt{3}}$$

Rationalize the denominator of \frac{8\sqrt{22}}{\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{8\sqrt{22}\sqrt{3}}{\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{8\sqrt{22}\sqrt{3}}{3}$$

To multiply \sqrt{22} and \sqrt{3}, multiply the numbers under the square root.

$$\frac{8\sqrt{66}}{3}$$