Cubes and Cube Roots — Mental Maths

Fill in the blanks:

(i) Cube of an even natural number is .....

(ii) Cubes of natural numbers are called .....

(iii) Cube of a natural number whose unit digit is 7, ends with digit .....

(iv) If a natural number has one zero at its end, then its cube will contain ..... zeros at the end.

(v) Finding cube root is the inverse operation of finding .....

(vi) If a number can be expressed as the triplets of equal prime factors, then it is called .....

Answer

(i) Cube of an even natural number is an even number.

(ii) Cubes of natural numbers are called perfect cubes.

(iii) Cube of a natural number whose unit digit is 7, ends with digit 3.

(iv) If a natural number has one zero at its end, then its cube will contain three zeros at the end.

(v) Finding cube root is the inverse operation of finding cube.

(vi) If a number can be expressed as the triplets of equal prime factors, then it is called perfect cube.

State whether the following statement is true (T) or false (F):

(i) Cube of any odd number is even.

(ii) A perfect cube does not end with two zeros.

(iii) There is no perfect cube which ends with 8.

(iv) The cube of a two digit number may be a three digit number.

(v) The cube of a single digit number may be a single digit number.

(vi) If a number is multiplied by 2 then its cube is multiplied by 8.

Answer

(i) False

Reason — Cubes of odd numbers are always odd. For example, 3³ = 27 and 5³ = 125, which are odd numbers.

(ii) True

Reason — If a number has one zero at its end, then its cube has three zeros at the end. So, the number of zeros at the end of a perfect cube is always a multiple of 3 and hence a perfect cube can never end with exactly two zeros.

(iii) False

Reason — If a number has 2 in the unit place, then its cube ends in 8. For example, 2³ = 8 and 12³ = 1728, which end with 8.

(iv) False

Reason — The smallest two digit number is 10 and 10³ = 1000, which is a four digit number. So, the cube of a two digit number always has four or more digits.

(v) True

Reason — 1³ = 1 and 2³ = 8, which are single digit numbers.

(vi) True

Reason — Let the number be $x$. On multiplying it by 2, we get $2x$.

$\Rightarrow (2x)^3\\[1em] \Rightarrow 2^3 \times x^3\\[1em] \Rightarrow 8x^3$

So, the cube of the number gets multiplied by 8.