If 2160 = 2a × 3b × 5c, find the value of 3a × 2−b × 5−c.

Concept

In a prime factorization, matching equal prime bases determines their exponents. For any non-zero base x, x−n = 1/xn; products of powers are simplified by evaluating each base separately.

Application

1. Factorize 2160: 2160 = 216 × 10 = (23 × 33) × (2 × 5) = 24 × 33 × 5. Therefore, a = 4, b = 3, and c = 1.

2. Substitute the exponents: 3a × 2−b × 5−c = 34 × 2−3 × 5−1.

3. Apply the negative-exponent rule: 34 × 1/23 × 1/5 = 81 × 1/8 × 1/5.

4. Multiply the factors: 81/(8 × 5) = 81/40.

Cross-check

Multiplying the obtained value by 23 × 5 gives (81/40) × 40 = 81 = 34, so the reciprocal factors have been handled consistently.

Result

Therefore, the required value is 81/40.