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\frac{2\times 4\sqrt{3}-3\sqrt{27}}{\sqrt{\frac{6\times 2+7}{2}}}
Factor 48=4^{2}\times 3. Rewrite the square root of the product \sqrt{4^{2}\times 3} as the product of square roots \sqrt{4^{2}}\sqrt{3}. Take the square root of 4^{2}.
\frac{8\sqrt{3}-3\sqrt{27}}{\sqrt{\frac{6\times 2+7}{2}}}
Multiply 2 and 4 to get 8.
\frac{8\sqrt{3}-3\times 3\sqrt{3}}{\sqrt{\frac{6\times 2+7}{2}}}
Factor 27=3^{2}\times 3. Rewrite the square root of the product \sqrt{3^{2}\times 3} as the product of square roots \sqrt{3^{2}}\sqrt{3}. Take the square root of 3^{2}.
\frac{8\sqrt{3}-9\sqrt{3}}{\sqrt{\frac{6\times 2+7}{2}}}
Multiply -3 and 3 to get -9.
\frac{-\sqrt{3}}{\sqrt{\frac{6\times 2+7}{2}}}
Combine 8\sqrt{3} and -9\sqrt{3} to get -\sqrt{3}.
\frac{-\sqrt{3}}{\sqrt{\frac{12+7}{2}}}
Multiply 6 and 2 to get 12.
\frac{-\sqrt{3}}{\sqrt{\frac{19}{2}}}
Add 12 and 7 to get 19.
\frac{-\sqrt{3}}{\frac{\sqrt{19}}{\sqrt{2}}}
Rewrite the square root of the division \sqrt{\frac{19}{2}} as the division of square roots \frac{\sqrt{19}}{\sqrt{2}}.
\frac{-\sqrt{3}}{\frac{\sqrt{19}\sqrt{2}}{\left(\sqrt{2}\right)^{2}}}
Rationalize the denominator of \frac{\sqrt{19}}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.
\frac{-\sqrt{3}}{\frac{\sqrt{19}\sqrt{2}}{2}}
The square of \sqrt{2} is 2.
\frac{-\sqrt{3}}{\frac{\sqrt{38}}{2}}
To multiply \sqrt{19} and \sqrt{2}, multiply the numbers under the square root.
\frac{-\sqrt{3}\times 2}{\sqrt{38}}
Divide -\sqrt{3} by \frac{\sqrt{38}}{2} by multiplying -\sqrt{3} by the reciprocal of \frac{\sqrt{38}}{2}.
\frac{-\sqrt{3}\times 2\sqrt{38}}{\left(\sqrt{38}\right)^{2}}
Rationalize the denominator of \frac{-\sqrt{3}\times 2}{\sqrt{38}} by multiplying numerator and denominator by \sqrt{38}.
\frac{-\sqrt{3}\times 2\sqrt{38}}{38}
The square of \sqrt{38} is 38.
\frac{-2\sqrt{3}\sqrt{38}}{38}
Multiply -1 and 2 to get -2.
\frac{-2\sqrt{114}}{38}
To multiply \sqrt{3} and \sqrt{38}, multiply the numbers under the square root.
-\frac{1}{19}\sqrt{114}
Divide -2\sqrt{114} by 38 to get -\frac{1}{19}\sqrt{114}.