Smallest Or Biggest Number amongst Exponential Numbers

 Q) Prelims 2022 --> Which number among  2⁴⁰ , 3²¹ , 4¹⁸ , and 8¹² is smallest ?

a) 2⁴⁰

b) 3²¹

c) 4¹⁸

d) 8¹²

Solution :- 

Category:- Quantitative Aptitude --> Basic Concept --> Laws of Indices

Common Strategies :- 

    1) Common Base

    2) Common Exponent 

Solving by both Common Exponent method and Common Base method

HCF of 21 , 18 and 12 is 3

-->  21 can be rewritten as 3 x 7 , hence, (3⁷)³= (3x3x3x3x3x3x3)³= (2187)³

-->  18 can be rewritten as 3 x 6 , hence, (4⁶)³= (4x4x4x4x4x4)³= (4096)³

-->  12 can be rewritten as 3 x 4 , hence, (8⁴)³= (8x8x8x8)³= (4096)³

2⁴⁰ can be written as 2 x (2³)¹³= 2 x 8¹³ by common base method it is greater than 8¹²  Hence Out of Option

so from above three Exponents the smallest value is (2187)³ Hence correct option is (b)