Find the cube root of each of the following numbers by prime factorisation method.
64
(i) By prime factorisation, we have
∴ 2 x 2
= 4
Thyus, cube root of 64 is 4.
Sides of the cubiod are 5 cm, 2 cm, 5 cm
∴ Volume of the cubiod = 5 cm x 2 cm x 5 cm
To form is as a cube its dimension should be in the group of triples.
∴ Volume of the required cube = [5 cm x 5 cm x 2 cm] x 5 cm x 2 cm x 2 cm
= [ 5 x 5 x 2 cm3 ] x 20 cm3
Thus, the required number of cubiods = 20.
Find the smallest number by which each of the following numbers must be divided to obtain a perfect cube. (i) 81 (ii) 128 (iii) 135 (iv) 192 (v) 704
(i) We have 81 = 3 x 3 x 3 x 3
Grouping the prime factors of 81 into triples, we are left with 3.
∴ 81 is not a perfect cube
Now, [81] 3= [3 x 3 x 3 x 3] 3
or 27 = 3 x 3 x 3
i.e. 27 is a perfect cube
Thus, the required smallest number is 3
(ii) we have 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2
Grouping the prime factors of 128 into triples, we are left with 2
∴ 128 is not a perfect cube
Now, [128] 2 = [2 x 2 x 2 x 2 x 2 x 2 x 2]2
or 64 = 2 x 2 x 2 x 2 x 2 x 2
i.e. 64 is a perfect cube
∴ the smallest required number is 2.
(iii) we have 135 = 3 x 3 x 3 x 5
Grouping the prime factors of 135 into triples, we are left over with 5.
∴ 135 is not a perfect cube
Now, [135]5 = [ 3 x 3 x 3 x 5] 5
or 27 = 3 x 3 x 3
i.e. 27 is a perfect cube.
Thus, the required smallest number is 5
(iv) We have 192 = 2 x 2 x 2 x 2 x 2 x 2 x 3
Grouping the prime factors of 192 into triples, 3 is left over.
∴ 192 is not a perfect cube.
Now, [192] 3= [2 x 2 x 2 x 2 x 2 x 2 x 3]3
or 64 = 2 x 2 x 2 x 2 x 2 x 2
i.e. 64 is a perfect cube.
Thus, the required smallest number is 3.
(v) We have 704 = 2 x 2 x 2 x 2 x 2 x 2 x 11
Grouping the prime factors of 704 into triples, 11 is left over
∴ [704]11 = [2 x 2 x 2 x 2 x 2 x 2 x 11]11
or 64 = 2 x 2 x 2 x 2 x 2 x 2
i.e. 64 is a perfect cube
Thus, the required smallest number is 11.
Find the smallest number by which each of the following numbers must be multiplied to obtain a perfect cube. (i) 243 (ii) 256 (iii) 72 (iv) 675 (v) 100
(i) We have 243 = 3 x 3 x 3 x 3 x 3
The prime factor 3 is not a group of three.
∴ 243 is not a perfect cube.
Now, [243] x 3 = [3 x 3 x 3 x 3 x 3] x 3
or, 729, = 3 x 3 x 3 x 3 x 3 x 3
Now, 729 becomes a [perfect cube
Thus, the smallest required number to multipkly 243 to make it a perfect cube is 3.
(ii) We have 256 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
Grouping the prime factors of 256 in triples, we are left over with 2 x 2.
∴ 256 is not a perfect cube.
Now, [256] x 2 = [2 x 2 x 2 x 2 x 2 x 2 x 2 x 2] x 2
or, 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2
i.e. 512 is a perfect cube.
thus, the required smallest number is 2.
(iii) we have 72 = 2 x 2 x 2 x 3 x 3
Grouping the prime factors of 72 in triples, we are left over with 3 x 3
∴ 72 is not a perfect cube.
Now, [72] x 3 = [2 x 2 x 2 x 3 x 3] x 3
or, 216 = 2 x 2 x 2 x 3 x 3 x 3
i.e. 216 is a perfect cube
∴ The smallest number required to multiply 72 to make it a perfect cube is 3.
(iv) We have 675 = 3 x 3 x 3 x 5 x 5
Grouping the prime factors of 675 to triples, we are left over with 5 x 5
∴ 675 is not a perfect cube.
Now, [675] x 5 = [3 x 3 x 3 x 5 x 5] x 5
Now, 3375 is a perfect cube
Thus, the smallest required number to multiply 675 such that the new number is a perfect cube is 5.
(v) We have 100 = 2 x 2 x 5 x 5
The prime factor are not in the groups of triples.
∴ 100 is not a perfect cube.
Now, [100] x 2 x 5 = [2 x 2 x 5 x 5] x 2 x 5
or, [100] x 10 = 2 x 2 x 2 x 5 x 5 x 5
1000 = 2 x 2 x 2 x 5 x 5 x 5
Now, 1000 is a perfect cube
Thus, the required smallest number is 10