Exercise 6.5

1. Find the cube root of the following numbers by prime factorization method.

(i) 343

Solution:

343 = × × = 7372

3343 = 373 =

(ii) 729

Solution:

729 = 3 × 3 × 3 × 3 × 3 × 3

= 3635 = 323 = 9394

3729 = 393 =

(iii) 1331

Solution:

1331 = 11 × 11 × 11 = 113123

31331 = 3113 =

(iv) 2744

Solution:

2744 = 2 × 2 × 2 × 7 × 7 × 7 = 2333 × 7393

= 143133

32744 = 3143 =

2. Find the cube root of the following numbers through estimation?

(i) 512

Solution:

512 is a perfect cube. 3512 =

(ii) 2197

Solution:

2197 is a perfect cube. 32197 =

(iii) 3375

Solution:

3375 is a perfect cube. 33375 =

(iv) 5832

Solution:

5832 is a perfect cube. 35832 =

3. State true or false?

(i) Cube of an even number is an odd number

Solution:

.

(ii) A perfect cube may end with two zeros

Solution:

.

A perfect cube must end with a multiple of zeros.

(iii) If a number ends with 5, then its cube ends with 5

Solution:

.

(iv) Cube of a number ending with zero has three zeros at its right

Solution:

.

(v) The cube of a single digit number is a single digit number.

Solution:

.

(vi) There is no perfect cube which ends with 8

Solution:

.

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

Solution:

.

The cube of a two-digit number will have at least digits and at most digits.

4. Find the two digit number which is a square number and also a cubic number.

Solution:

We are looking for a number x such that x = a2 and x = b3 where x is a two-digit number.

This means x must be a power (x = c6).

Let's check sixth powers:

16 =

26 =

36 =

Therefore, the two-digit number is 64.