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Example 9. Find the cube root of 238328. = 21 + 23 + 25 + 27 + 29
Solution: 238328 is an even number so its cube and so on, what do we observe from the above
root has to an even number. pattern?
Make two groups ® 238 and 328 We see that cube of a number ‘n’ can be
First group 328 ® unit digit is 8 so unit expressed as the sum of the ‘n’ odd
digit of cube root will be 2. consecutive number.
(see the above table). 2. Let us see another pattern
3
3
3
Second Group 238 ® it lies between 6 = 2 - 1 = + 2 ´ ´ 3
1
1
3
216 and 7 = 343 . 3 - 2 = + 3 ´ 2 ´ 3
3
3
1
So take the smaller number 6 as ten’s place. 4 - 3 = + 4 ´ 3 ´ 3
3
3
1
\ 3 238328 = 62 5 - 4 = + 5 ´ 4 ´ 3
3
3
1
3
3
Example 10. Find the cube root of: 175616. 6 - 5 = +1 6 ´ 5 ´ 3 and so on.
Solution: We know that the number is even so 3. Now let us look at this pattern
its cube root has to an even number. 1 - 0 =1
3
3
Make two groups ® 175 and 616 2 - 1 = 8 - = 7
3
3
1
3
3
First group 616 ® unit digit is 6 so unit 3 - 2 = 27 - 8 19
=
digit of cube root will be 6 4 - 3 = 64 - 27 = 37
3
3
(see the above table). 3 3
5 - 4 =125 - 64 = 61
3
Second Group 175 ® it lies between 5 = 3 3
-
3
125 and 6 = 216 . 6 - 5 = 216 125 = 91
3
\ 1 =1
So take the smaller number 5 as ten’s place.
3
2 = + 7
1
3 175616 = 56
3
1
3 = + 7 + 19
3
Some Patterns involving Cubic Numbers 4 = +1 7 + 19 + 37
3
Let us look at some interesting patterns 5 = +1 7 + 19 + 37 + 61
3
+
+
+
expressing cubic numbers 6 = + +1 7 19 37 61 91
3
1
1
1 = ´ ´ =1 and so on.
1
=1 Thus, we observe that
3
3
2 = 2 ´ 2 ´ 2 = 8 1 is sum of 1
3
= 3 + 5 2 is sum of 1, 7
3
3
3 = 3 ´ 3 ´ 3 = 27 3 is sum of 1, 7, 19
3
= 7 + 9 + 11 4 is the sum of 1, 7, 19 37 and so on.
3
4 = 4 ´ 4 ´ 4 = 64 The last number in each case, that is (1, 7,
=13 + 15 + 17 + 19 19, 37, 61) may be obtained by putting n = 0,
1, 2, 3, 4 ........... in [1 + n (n + 1) ´ 3]
3
5 = 5 ´ 5 ´ 5 125
=
E xercise 4.3
1. State the unit digits of the cube root of following cubed numbers.
(i) 343 (ii) 1728 (iii) 1331 (iv) 2744
2. Find the cube root of following cube numbers by estimation.
(i) 9261 (ii) 42875 (iii) 17576 (iv) 32768
(v) 205379 (vi) 35937
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Mathematics In Focus - 8