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\[P = \left( {\frac{{x - 108 + 23\sqrt x }}{{x - 16}} - 1} \right)\,:\,\left( {\frac{{75 - x}}{{x + \sqrt x - 12}} + \frac{{\sqrt x + 3}}{{\sqrt x + 4}}} \right)\,,\,\left( {x > 0,\,x \ne 9,\,x \ne 16} \right)\].

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Giải thích

Điều kiện \[\left\{ \begin{array}{l}x > 0\\x \ne 9\\x \ne 16\end{array} \right.\].

Ta có \[P = \left( {\frac{{x - 108 + 23\sqrt x }}{{x - 16}} - 1} \right)\,:\,\left( {\frac{{75 - x}}{{x + \sqrt x - 12}} + \frac{{\sqrt x + 3}}{{\sqrt x + 4}}} \right)\,\]

\[ = \left( {\frac{{x - 16 + 23\sqrt x - 92}}{{x - 16}} - 1} \right)\,:\,\left[ {\frac{{\left( {75 - x} \right)\left( {\sqrt x + 4} \right) + \left( {\sqrt x + 3} \right)\left( {x + \sqrt x - 12} \right)}}{{\left( {\sqrt x + 4} \right)\left( {x + \sqrt x - 12} \right)}}} \right]\,\,\]

\[ = \frac{{23\left( {\sqrt x - 4} \right)}}{{{{\left( {\sqrt x } \right)}^2} - 16}}\,:\,\,\left[ {\frac{{75\sqrt x + 300 - x\sqrt x - 4x + x\sqrt x + x - 12\sqrt x + 3x + 3\sqrt x - 36}}{{\left( {\sqrt x + 4} \right)\left( {x + \sqrt x - 12} \right)}}} \right]\,\]

\[ = \frac{{23}}{{\sqrt x + 4}}\,:\,\left[ {\frac{{66\sqrt x + 264}}{{\left( {\sqrt x + 4} \right)\left( {x + \sqrt x - 12} \right)}}} \right]\]\[ = \frac{{23\left( {x - 16 + \sqrt x + 4} \right)}}{{66\left( {\sqrt x + 4} \right)}}\, = \frac{{23}}{{66}}\left( {\sqrt x - 3} \right)\]