Short Answer TypeIs 1372 a perfect cube? If not, find the smallest natural number by which 1372 must be multipled so that the product is a perfect cube.
Find the one’s digit of the cube of each of the following numbers.
(i) 3331 (ii) 8888 (iii) 149 (iv) 1005
(v) 1024 (vi) 77 (vii) 5022 (viii) 53
Consider the following pattern:
(a) 63 (b) 83 (c) 73
Long Answer TypeWhich of the following are perfect cubes?
1. 400 2. 3375 3. 8000 4. 15625
5. 9000 6. 6859 7. 2025 8. 10648
Check which of the following are perfect cubes.
(i) 2700 (ii) 16000 (iii) 64000 (iv) 900
(v) 125000 (vi) 36000 (vii) 21600 (viii) 10000
(ix) 27000000 (x) 1000
What pattern do you observe in these perfect cubes?
Short Answer TypeWhich of the following numbers are not perfect cubes? (i) 216 (ii) 128 (iii) 1000 (iv) 100 (v) 46656
Long Answer TypeFind 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
Short Answer Type
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