Which number amongst 2⁴⁰, 3²¹, 4¹⁸ and 8¹² is the smallest?
- (a) 2⁴⁰
- (b) 3²¹ ✓ UPSC's answer
- (c) 4¹⁸
- (d) 8¹²
Why the answer is (b)
• Write 4¹⁸ and 8¹² as powers of 2: 4¹⁸ = (2²)¹⁸ = 2³⁶ and 8¹² = (2³)¹² = 2³⁶, so both equal 2³⁶.
• 2⁴⁰ is larger than 2³⁶, so it cannot be the smallest.
• Compare 3²¹ with 2³⁶ by taking a common exponent of 3: 3²¹ = (3⁷)³ = 2187³ and 2³⁶ = (2¹²)³ = 4096³.
• Since 2187 < 4096, 3²¹ < 2³⁶ < 2⁴⁰.
• So 3²¹ is the smallest, option (b).
Why the other options are wrong
- (a) 2⁴⁰
- 2⁴⁰ is the largest of the four numbers, since 4¹⁸ and 8¹² both equal 2³⁶ and 3²¹ is smaller still.
- (c) 4¹⁸
- 4¹⁸ = 2³⁶ = 4096³, which is larger than 3²¹ = 2187³.
- (d) 8¹²
- 8¹² = 2³⁶ = 4096³, which is larger than 3²¹ = 2187³.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2022, held on 5 June 2022. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.