45.

You are told that 1,331 is a perfect cube. Can you guess without factorisation what is its cube root? Similarly, guess the cube roots of 4913, 12167, 32768.

(i) Separating the given number (1331) into two groups :

1331  1 and 331
∵ 331 end in 1
∴ Unit's digit of the cube root = 1

∵ 1³ = 1 and 
∴ Ten's digit of the cube root = 1

∴ 

(ii)  Separating the given number (4913) in two groups:

4913 4 and 913
Unit's digit:
∵ Unit's digit in 913  is 3
∴ Unit's digit of the cube root = 7

[7³ = 343 : which ends in 3]

Ten's digit:
∵ 1³ = 1, 2³  = 8
and 1 < 4 < 8
i.e. 1³  <  4 < 2³ 
∴ Then ten's digit of the cube root is 1.

∴  

(iii) Separatibng 12167 in two groups:
     1216712 and 167
Unit's digit :
∵ 167 is ending in 7 and cube of a number ending in 3 ends in 7
∴ The unit's digit of the cube root = 3
Ten's digit
∵ 2³  = 8 adn 3³ = 27
Also,  8 < 12 < 27
or,  2³ < 12 < 3²  

∴ The tens digit of the cube root can be 2.

Thus,   

(iv) separating 32768 in two groups:

32768 32 and 786
Unit's digit:
768 will guess the unit's digit in the cube root.
∵ 768 ends in 8.

∴ Unit's digit in the cube root = 2

Ten's digit:
∵ 3³ = 27 and 4³ - 64
Also, 27 < 32 < 64
or, 3³ < 32 < 4³ 

∴ The ten's digit of the cube root = 3

Thus,