What is the value of x, if (2/7)-8 × (7/2)-2 = (2/7)2x ?

Concept: For a nonzero base a (a ≠ 1, -1, 0), the negative-exponent rule gives a-n = (1/a)n, and same-base powers multiply by adding exponents: am × an = am+n. If am = an for such a base, then m = n.

1. Rewrite (7/2)-2 using the negative-exponent reciprocal rule: (7/2)-2 = (2/7)2.

2. Substitute this back into the equation: (2/7)-8 × (2/7)2 = (2/7)2x.

3. Both terms on the left now share the base 2/7, so add the exponents: -8 + 2 = -6, giving (2/7)-6 = (2/7)2x.

4. Since the bases on both sides are equal (and not 0, 1, or -1), the exponents must be equal: 2x = -6.

5. Solve for x: x = -3.

Cross-check: substituting x = -3 back, 2x = -6, so the right-hand side is (2/7)-6, which matches the left-hand side value obtained above — confirming the solution is consistent.