If m and n are whole numbers with n > 1 such that mn = 121, then the value of…

Concept

A relation of the form ab = N is solved by writing N as a perfect power of a smaller base. Factor N into primes: 121 = 112. When the prime-power exponent is itself a prime (here 2), that factorisation is the ONLY way to write N as a perfect power with an exponent greater than 1, so the condition on the exponent fixes the base and exponent uniquely. With a and b fixed, any target expression in a and b is found by direct substitution.

Application

1. Express 121 as a prime power.

   • 121 = 11 × 11 = 112.

   • The condition n > 1 rules out the trivial 1211 reading, so matching mn = 112 gives the unique whole numbers m = 11 and n = 2.

2. Substitute into the target expression.

   • Base: m − 1 = 11 − 1 = 10.

   • Exponent: n + 1 = 2 + 1 = 3.

   • So (m − 1)n+1 = 103.

3. Compute.

   • 103 = 10 × 10 × 10 = 1000.

Cross-check

Verify the base condition: with m = 11, n = 2, mn = 112 = 121, which matches the given relation, and n > 1 holds. Hence (m − 1)n+1 = 1000.