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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

If the operator ∇ is defined as:

equation

which one(s) of the following choices is/are not equivalent to:

equation

First of all, the expression is the product of two powers:

equation

which eliminates choices II and IV. Choice IV is obviously incorrect since it assumes a ternary operator, whereas ∇ is a binary operator.

We must still verify the accuracy of both other expressions:

equation

equation

equation

therefore, expression III is correct.

equation

equation

equation

and expression I is also correct.