Find a number that can replace y in the expression (x4)0 = x2/3 · xy, where x…

For a base a (a ≠ 0, ±1), two exponent rules apply: (am)n = amn (power of a power), and am · an = am+n (product of powers). Also, if two expressions with the same base are equal, ap = aq, then the exponents themselves must be equal: p = q. These facts convert an exponential equation into a plain algebraic equation in the exponents.

1. Simplify the left side using the power-of-a-power rule: (x4)0 = x4×0 = x0.

2. Simplify the right side using the product rule: x2/3 · xy = x2/3+y.

3. Since both sides now have the same base x and are equal, equate the exponents: 0 = 2/3 + y.

4. Solve this linear equation for y: y = -2/3.

Cross-check: substituting y = -2/3 back into the right side gives x2/3 · x-2/3 = x2/3-2/3 = x0, which matches the simplified left side, x0, exactly — confirming y = -2/3.