Which number amongst 2⁴⁰, 3²¹, 4¹⁸ and 8¹² is the smallest?

Answer: (b) 3²¹

4¹⁸ and 8¹² both equal 2³⁶, which is smaller than 2⁴⁰. And 3²¹ = 2187³ is smaller than 2³⁶ = 4096³, so 3²¹ is the smallest of the four.

1. Write the powers of 2 in base 2: 4¹⁸ = 2³⁶ and 8¹² = 2³⁶. Among 2⁴⁰, 4¹⁸ and 8¹² the smallest is 2³⁶.
2. Compare 3²¹ with 2³⁶ using the common exponent 3: 3²¹ = (3⁷)³ = 2187³ and 2³⁶ = (2¹²)³ = 4096³.
3. 2187 < 4096, so 3²¹ < 2³⁶ < 2⁴⁰.
4. Check: 21 × log 3 ≈ 10.02 while 36 × log 2 ≈ 10.84, so 3²¹ (about 1.05 × 10¹⁰) is the smallest.

Remember · To compare powers, rewrite them with a common base or a common exponent, then compare only what differs.

Question and answer: UPSC's official GS Paper II (2022, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). · Report an error