Page 84 - Start Up Mathematics_8 (Non CCE)

Example 15: Find the cube roots of the following rational numbers by prime factorization:

343 -512
(a) (b)
1 331, -3 375,

343 3 343
Solution: (a) 3 =
,
1 331 3 1 331
,
Writing 343 and 1,331 as a product of
their prime factors, we get 343 = 7 × 7 × 7 7 343
7 49
and 1,331 = 11 × 11 × 11 7 7
1
\ 343 = 3 7 ¥ ¥ 7 and 3 1 331 =, 3 11 11 11¥ ¥
3
7
= 7 = 11 11 1331
343 3 343 7 11 121
\ 3 = =
,
1 331 3 1 331 11 11 11
,
1
-512 3 -512 - 512 3 512
3
(b) 3 = = =
-3 375 3 -3 375 - 3 375 3 3 375
,
,
,
,
3
Writing 512 and 3,375 as a product of their prime factors, we get
2 512
512 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 2 256
and 3,375 = 5 × 5 × 5 × 3 × 3 × 3 5 3,375 2 128
3 512 = 2 2 2 2 2 2 2 2 2 5 675 2 64
¥
¥
¥
¥
¥
¥
¥
¥
3
5 135 2 32
= 2 × 2 × 2 = 8 3 27 2 16
3 3 375, = 5 5 5 3 3 3 = 5 × 3 = 15 3 9 2 8
¥
¥
¥
¥
¥
3
2
4
3 3
-512 3 512 8 1 2 2
\ 3 = = 1
,
-3 375 3 3 375 15
,
Example 16: Find the cube roots: (a) 2.744 (b) –0.004913
2 744, 3 2 744,
3
Solution: (a) 2 744. = 3 =
1 000, 3 1 000,
2,744 = 2 ¥ 2 ¥ 2 ¥ 7 ¥ 7 ¥ 7
2 2,744
\ 2 744, = 3 2 ¥¥¥¥¥ 7 2 1,372
2
7
7
2
3
= 2 × 7 = 14 2 686
1,000 = 10 × 10 × 10 7 343
7 49
\ 1 000, = 3 10 10 10¥ ¥ = 10 7 7
3
1
3 2 744, 14
\ 3 2 744. = = = 14.
3 1 000, 10
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