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If $72^{\mathrm{x}} \cdot 48^{\mathrm{y}}=6^{\mathrm{x} y}$, where $\mathrm{x}$ and $\mathrm{y}$ are nonzero rational numbers, then $\mathrm{x}+\mathrm{y}$ equals
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Verified Answer
The correct answer is:
$-\frac{10}{3}$
$72^{\mathrm{x}} \cdot 48^{\mathrm{y}}=6^{\mathrm{xy}}$
$3^{2 \mathrm{x}+y} \cdot 2^{3 \mathrm{x}+4 y}=2^{\mathrm{xy}} \cdot 3^{\mathrm{xy}}$
compare
$2 x+y=x y$ .........(i)
$\& 3 x+4 y=x y$ .........(ii)
From (i) $\&$ (ii) $\quad x=-3 y$
put in (i)
$-5 y=-3 y^{2}$
$\Rightarrow y=\frac{5}{3}$
so $x+y=-2 y=-\frac{10}{3}$
$3^{2 \mathrm{x}+y} \cdot 2^{3 \mathrm{x}+4 y}=2^{\mathrm{xy}} \cdot 3^{\mathrm{xy}}$
compare
$2 x+y=x y$ .........(i)
$\& 3 x+4 y=x y$ .........(ii)
From (i) $\&$ (ii) $\quad x=-3 y$
put in (i)
$-5 y=-3 y^{2}$
$\Rightarrow y=\frac{5}{3}$
so $x+y=-2 y=-\frac{10}{3}$
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