Cube And Cube Roots Class 8
Cube And Cube Roots Class 9
Exercise:
Q 1. Examine which of the following numbers are perfect cubes. In case of perfect cubes, Find the cube root:
i) 729 =
3 | 729
3 | 243
3 | 81
3 | 27
3 | 9
3 | 3
| 1
Resolving 729 into prime factors, we get :
729 = 3 * 3 * 3 * 3 * 3 * 3
∛729 = 3 * 3 = 9
ii) 1331 =
11 | 1331
11 | 121
11 | 11
| 1
Resolving 1331 into prime factors, we get :
1331 = 11* 11 * 11
∛1331 = 11 = 11
iii) 5324 :
2 | 5324
2 | 2662
11 | 1331
11 | 121
11 | 11
| 1
Resolving 5324 into prime factors, we get :
5324 = 2 * 2 * 11 * 11 * 11
Clearly, 5324 is not a Perfect cube.
iv) 3375 :
3 | 3375
3 | 1125
3 | 375
5 | 125
5 | 25
5 | 5
| 1
Resolving 3375 into prime factors, we get :
3375 = 3 * 3 * 3 * 5 * 5 * 5
∛3375 = 3 * 5 = 15
v) 9261 :
3 | 9261
3 | 3087
3 | 1029
7 | 343
7 | 49
7 | 7
| 1
Resolving 9261 into prime factors, we get :
9261 = 3 * 3 * 3 * 7 * 7 * 7
∛9261 = 3 * 7 = 21
vi) 5832 :
3 | 5832
3 | 1944
3 | 648
3 | 216
3 | 72
3 | 24
2 | 8
2 | 4
2 | 2
| 1
Resolving 5832 into prime factors, we get :
5832 = 3 * 3 * 3 * 3 * 3 * 3 * 2 * 2 * 2
∛5832 = 3 * 3 * 2 = 18
vii) 1728 :
3 | 1728
3 | 576
3 | 192
2 | 64
2 | 32
2 | 16
2 | 8
2 | 4
2 | 2
| 1
Resolving 1728 into prime factors, we get :
1728 = 3 * 3 * 3 * 2 * 2 * 2 * 2 * 2 * 2
∛1728 = 3 * 2 * 2 = 12
viii) 10584 :
2 | 10584
2 | 5292
2 | 2646
3 | 1323
3 | 441
3 | 147
7 | 49
7 | 7
| 1
Resolving 10584 into prime factors, we get :
10584 = 2 * 2 * 2 * 3 * 3 * 3 * 7 * 7
Clearly, 10584 is not a perfect cube.
Q. 2) Find the smallest number by which 1323 must be multiplied so that the product is a perfect cube.
Solution :
Resolving 1323 into prime factors we get
3 | 1323
3 | 441
3 | 147
7 | 49
7 | 7
| 1
1323 = 3 * 3 * 3 * 7 * 7
Clearly to make it a perfect cube it must be multiplied by 7.
Q. 3 ) What is the smallest number by which 1600 must be divided so that the quotient is the perfect cube.
Solution :
Resolving 1600 into prime factors we get
2 | 1600
2 | 800
2 | 400
2 | 200
2 | 100
2 | 50
5 | 25
5 | 5
| 1
1600 = 2 * 2 * 2 * 2 * 2 * 2 * 5 * 5
Clearly, to make it a perfect cue it must be divided by 5 * 5 i.e 25.
Q. 4) Find the cube root of :
i) 216
2197
Solution : 216 = 3 * 3 * 3 * 2 * 2 * 2
2197 13 * 13 * 13
∛216 = 3 * 2 = 6
∛2197 13 13
ii) 4 508 = 5832
1331 1331
Solution : 5832 = 2 * 2 * 2 * 3 * 3 * 3 * 3 * 3 * 3
1331 11 * 11 * 11
∛5832 = 2 * 3 * 3 = 18
∛1331 11 11
iii) 42.875 = 42875 = 5 * 5 * 5 * 7 * 7 * 7
1000 10 * 10 * 10
∛42875 = 5 * 7 = 35 = 3.5
∛1000 10 10
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