Answer:
- y = -48/x . . . . . infinite possibilities
- for integer solutions, see below
Step-by-step explanation:
The equation can be put into more conventional form:
![2\sqrt[3]{3xy}=-4\sqrt[3]{18} \qquad\text{given}\\\\\sqrt[3]{3xy}=-2\sqrt[3]{18} \qquad\text{divide by 2}\\\\3xy=-8\cdot 18 \qquad\text{cube both sides}\\\\y=\dfrac{-48}{x} \qquad\text{divide by $3x$}](https://tex.z-dn.net/?f=2%5Csqrt%5B3%5D%7B3xy%7D%3D-4%5Csqrt%5B3%5D%7B18%7D%20%5Cqquad%5Ctext%7Bgiven%7D%5C%5C%5C%5C%5Csqrt%5B3%5D%7B3xy%7D%3D-2%5Csqrt%5B3%5D%7B18%7D%20%5Cqquad%5Ctext%7Bdivide%20by%202%7D%5C%5C%5C%5C3xy%3D-8%5Ccdot%2018%20%5Cqquad%5Ctext%7Bcube%20both%20sides%7D%5C%5C%5C%5Cy%3D%5Cdfrac%7B-48%7D%7Bx%7D%20%5Cqquad%5Ctext%7Bdivide%20by%20%243x%24%7D)
This shows an inverse relationship between x and y. No specific values are indicated. There are an infinite number of possible pairs of values for x and y.
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Possible integer values are ...
(x, y) ∈ {(-1, 48), (-2, 24), (-3, 16), (-4, 12), (-6, 8), (-8, 6), (-12, 4), (-16, 3), (-24, 2), (-48, 1), (1, -48), (2, -24), (3, -16), (4, -12), (6, -8), (8, -6), (12, -4), (16, -3), (24, -2), (48, -1)}