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 $y = \frac{5}{3}$
$\therefore x + y = \frac{-15}{3} + \frac{5}{3} = \frac{-10}{3}$