PDF chapter test TRY NOW
1. The cube of a positive number is always positive.
Example:
4^3 = 4 \times 4 \times 4 = 64
2. The cube of a negative number is always negative.
Example:
(-4)^3 = (-4) \times (-4) \times (-4) = -64
3. The cube of every even number is even.
Example:
2^3 = 8, 4^3 = 64, 6^3 = 216, 8^3 = 512, ...
Here, 8, 64, 216 and 512 are all even numbers.
4. The cube of every odd number is odd.
Example:
1^3 = 1, 3^3 = 27, 5^3 = 125, 7^3 = 343, ...
Here, 1, 27, 2125 and 343 are all odd numbers.
5. If a natural number ends at 0, 1, 4, 5, 6 or 9, its cube also ends with the same 0, 1, 4, 5, 6 or 9, respectively.
Example:
(i) 10^3 = 100\underline{0}
(ii) 1^3 = \underline{1}
(iii) 4^3 = 6\underline{4}
(iv) 5^3 = 12\underline{5}
(v) 6^3 = 21\underline{6}
(vi) 9^3 = 72\underline{9}
6. If a natural number ends at 2 or 8, its cube ends at 8 or 2, respectively.
Example:
(i) 2^3 = \underline{8}
(ii) 8^3 = 51\underline{2}
7. If a natural number ends at 3 or 7, its cube ends at 7 or 3, respectively.
Example:
(i) 3^3 = 2\underline{7}
(ii) 7^3 = 34\underline{3}
8. A perfect cube does not end with two zeroes.
Example:
10^3 = 1000, 20^3 = 8000, …
9. The sum of the cubes of first n natural numbers is equal to the square of their sum.
That is, 1^3 + 2^3 + 3^3 + 4 ^3 + …. + n^3 = (1 + 2 + 3 + 4 + … + n)^2
Example:
1^3 + 2^3 + 3^3 = 1 + 8 + 27 = 36
(1 + 2 +3)^2 = 6^2 = 36
So, 1^3 + 2^3 + 3^3 = (1 + 2 +3)^2
10. Each prime factor of a number appears three times in its cube.
Example:
6^3 = 216
Prime factor of 6 = 2 \times 3
Prime factor of 216 = (2 \times 2 \times 2) \times (3 \times 3 \times 3)
11. There are only three numbers whose cube is equal to itself.
(i) 0^3 = 0 \times 0 \times 0 = 0
(ii) 1^3 = 1 \times 1 \times 1 = 1
(iii) (-1)^3 = (-1) \times (-1) \times (-1) = -1
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