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\frac{-3\left(8-\sqrt{3}\right)}{\left(8+\sqrt{3}\right)\left(8-\sqrt{3}\right)}
Rationalize the denominator of \frac{-3}{8+\sqrt{3}} by multiplying numerator and denominator by 8-\sqrt{3}.
\frac{-3\left(8-\sqrt{3}\right)}{8^{2}-\left(\sqrt{3}\right)^{2}}
Consider \left(8+\sqrt{3}\right)\left(8-\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.
\frac{-3\left(8-\sqrt{3}\right)}{64-3}
Square 8. Square \sqrt{3}.
\frac{-3\left(8-\sqrt{3}\right)}{61}
Subtract 3 from 64 to get 61.
\frac{-24+3\sqrt{3}}{61}
Use the distributive property to multiply -3 by 8-\sqrt{3}.