Find the value of 'x', 'y' and 'z' if: (45)9 × (36)3 × (20)2 = 2(2x − 2) ×…

Concept: When both sides of an equation are written as products of prime powers, factor every composite base into its prime factors, apply the power-of-a-product rule (ab)n = an bn and the power-of-a-power rule (am)n = amn, and combine terms with the same prime base by adding their exponents. Because prime factorization is unique, once both sides are expressed using the same primes, the exponents of each prime must be equal — this lets the equation be split into one equation per prime and solved for the unknowns.

Applying it here:

1. Factor each base into primes: 45 = 32 × 5, 36 = 22 × 32, and 20 = 22 × 5.

2. Substitute these into the left side: (32 × 5)9 × (22 × 32)3 × (22 × 5)2 = 2(2x − 2) × 3(5y + 4) × 5(z + 4).

3. Apply the power-of-a-power rule to each factor: 318 × 59 × 26 × 36 × 24 × 52 = 2(2x − 2) × 3(5y + 4) × 5(z + 4).

4. Combine the exponents of each matching prime: 2(6+4) × 3(18+6) × 5(9+2) = 210 × 324 × 511.

5. Equate the powers of 2: 2x − 2 = 10, so x = 6.

6. Equate the powers of 3: 5y + 4 = 24, so y = 4.

7. Equate the powers of 5: z + 4 = 11, so z = 7.

Cross-check: Substituting x = 6, y = 4, z = 7 back into the right side gives 2(2×6 − 2) × 3(5×4 + 4) × 5(7+4) = 210 × 324 × 511, which exactly matches the simplified left side — confirming the values independently.

∴ x = 6, y = 4, and z = 7.