If 2160 = 2^a.3^b.5^c, find a, b and c. Hence, calculate the value of 3^a × 2^-b × 5^-c.

Factorizing 2160, we get :

⇒ 2160 = 2⁴ × 3³ × 5¹

⇒ 2^a.3^b.5^c = 2⁴ × 3³ × 5¹

⇒ a = 4, b = 3 and c = 1.

Substituting values of a, b and c in 3^a × 2^-b × 5^-c, we get :

$\Rightarrow 3^4 \times 2^{-3} \times 5^{-1} = 81 \times \dfrac{1}{2^3} \times \dfrac{1}{5} \\[1em] = 81 \times \dfrac{1}{8} \times \dfrac{1}{5} \\[1em] = \dfrac{81}{40} = 2\dfrac{1}{40}.$

Hence, a = 4, b = 3 and c = 1 and 3^a × 2^-b × 5^-c = $2\dfrac{1}{40}$.