# Cubes And Cube Roots

## Exercise 6.2

1). Find the cube root of each of the following numbers by prime factorisation method.

(i) 64 (ii) 512

(iii) 10648 (iv) 27000

(v) 15625 (vi) 13824

(vii) 110592 (viii) 46656

(ix) 175616 (x) 91125

Solution:

(i) 64

64 = 2 X 2 X 2 X 2 X 2 X 2

64 = __2 X 2 X 2__ X __2 X 2 X 2__

64 = 2^{3} X 2^{3}

^{3}√64 = ^{3}√2^{3} X 2^{3}

= 2 X 2

= 4

Ans: ^{3}√64 = 4

(ii) 512

512 = 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2

512 = __2 X 2 X 2__ X __2 X 2 X 2 __X __2 X 2 X 2__

512 = 2^{3} X 2^{3 }X 2^{3}

^{3}√512 = ^{3}√2^{3} X 2^{3 }X 2^{3}

= 2 X 2 X 2

= 8

Ans: ^{3}√512 = 8

(iii) 10648

10648 = 2 X 2 X 2 X 11 X 11 X 11

10648 = __2 X 2 X 2__ X __11 X 11 X 11__

10648 = 2^{3} X 11^{3}

^{3}√10648 = ^{3}√2^{3} X 11^{3}

= 2 X 11

= 22

Ans: ^{3}√10648= 22

(iv) 27000

27000= 2 X 2 X 2 X 3 X 3 X 3 X 5 X 5 X 5

27000= __2 X 2 X 2__ X __3 X 3 X 3 __X __5 X 5 X 5__

27000= 2^{3} X 3^{3 }X 5^{3}

^{3}√27000 = ^{3}√2^{3} X 3^{3 }X 5^{3}

= 2 X 3 X 5

= 30

Ans: ^{3}√27000 = 30

(v) 15625

15625= 5 X 5 X 5 X 5 X 5 X 5

15625= __5 X 5 X 5__ X __5 X 5 X 5__

15625= 5^{3} X 5^{3}

^{3}√15625 = ^{3}√5^{3} X 5^{3}

= 5 X 5

= 25

Ans: ^{3}√15625 = 25

(vi) 13824

13824= 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 3 X 3 X 3

13824= __2 X 2 X 2__ X __2 X 2 X 2 __X __2 X 2 X 2__ X __3 X 3 X 3__

13824= 2^{3} X 2^{3 }X 2^{3} X 3^{3}

^{3}√13824 = ^{3}√2^{3} X 2^{3 }X 2^{3} X 3^{3}

= 2 X 2 X 2 X 3

= 24

Ans: ^{3}√13824 = 24

(vii) 110592

110592= 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2X 3 X 3 X 3

110592= __2 X 2 X 2__ X __2 X 2 X 2__ X __2 X 2 X 2__ X __2 X 2 X 2__X __3 X 3 X 3__

110592= 2^{3} X 2^{3} X 2^{3 }X 2^{3} X 3^{3}

^{3}√110592 = ^{3}√2^{3} X 2^{3} X 2^{3 }X 2^{3} X 3^{3}

= 2 X 2 X 2 X 2 X 3

= 48

Ans: ^{3}√110592 = 48

(viii) 46656

46656= 2 X 2 X 2 X 2 X 2 X 2 X 3 X 3 X 3 X 3 X 3 X 3

46656= __2 X 2 X 2__ X __2 X 2 X 2 __X __3 X 3 X 3__ X __3 X 3 X 3__

46656= 2^{3} X 2^{3} X 3^{3 }X 3^{3}

^{3}√46656 = ^{3}√2^{3} X 2^{3} X 3^{3 }X 3^{3}

= 2 X 2 X 3 X 3

= 36

Ans: ^{3}√46656 = 36

(ix) 175616

175616= 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 2 X 7 X 7 X 7

175616= __2 X 2 X 2__ X __2 X 2 X 2 __X __2 X 2 X 2__ X __7 X 7 X 7__

175616= 2^{3} X 2^{3 }X 2^{3 }X 7^{3}

^{3}√175616 = ^{3}√2^{3} X 2^{3 }X 2^{3 }X 7^{3}

= 2 X 2 X 2 X 7

= 56

Ans: ^{3}√175616 = 56

(x) 91125

91125= 3 X 3 X 3 X 3 X 3 X 3 X 5 X 5 X 5

91125= __3 X 3 X 3__ X __3 X 3 X 3__ X __5 X 5 X 5__

91125= 3^{3} X 3^{3 }X 5^{3}

^{3}√91125 = ^{3}√3^{3} X 3^{3 }X 5^{3}

= 3 X 3 X 5

= 45

Ans: ^{3}√91125 = 45

2). State true or false.

i). Cube of any odd number is even.

Ans: False

ii). A perfect cube does not end with two zeroes.

Ans: True

iii). If square of a number ends with 5, then its cube ends with 25.

Ans: True

iv). There is no perfect cube which ends with 8.

Ans: False

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

Ans: False

vi). The cube of a two digit number may have seven or more digits.

Ans: False

vii). The cube of a single digit number may be a single digit number.

Ans: False

Click here for the solutions of Std 8 Maths

1). Rational Numbers

2). Linear Equations in One Variable

3). Understanding Quadrilaterals

4). Data Handling

5). Squares and Square Roots

6). Cubes and Cube Roots