Question

If $72^x\cdot 48^y=6^{xy}$, where $x$ and $y$ are non-zero rational numbers, then $x+y$ equals

• A. $3$
• B. $\frac{10}{3}$
• C. $-3$
• D. $-\frac{10}{3}$

Answer 1

Answer: (D)

Given, $72^x\cdot 48^y=6^{xy}$
$(2^3\cdot 3^2{)}^x\cdot (2^4\cdot 3{)}^y=2^{xy}\cdot 3^{xy}$
$2^{3x+4y}\cdot 3^{2x+y}=2^{xy}\cdot 3^{xy}$
Equating the exponent of $2$ and $3$, we get
$3x+4y=xy$ and $2x+y=xy$
On solving these equation, we get
$x=\frac{-15}{3}$ and $y=\frac{5}{3}$
$\therefore x+y=\frac{-15}{3}+\frac{5}{3}=\frac{-10}{3}$