Page 77 - Start Up Mathematics_8 (Non CCE)

9. Write the cubes of 7 natural numbers which are of the form 3n + 2 (i.e., 5, 8, 11, 14, ...) and verify
that “The cube of a natural number of the form 3n + 2 when divided by 3 leaves remainder 2”.
10. Find the smallest number by which the following numbers must be multiplied so that the products
are perfect cubes.
(a) 5,324 (b) 1,323 (c) 27,783 (d) 12,348 (e) 3,125 (f) 1,01,306
11. Find the volume of a cubical box whose edge is 13 cm.
12. Find the smallest number by which the following should be divided to get the quotient as a perfect
cube.
(a) 1,536 (b) 9,826 (c) 3,97,535 (d) 3,26,592 (e) 3,31,776 (f) 8,788
13. Prove that if a number is tripled, then its cube is 27 times the cube of the given number.
14. What happens to the cube of a number if the number is multiplied by the following?
(a) 2 (b) 7
15. Find the cubes of the following using the column method.
(a) 63 (b) 29 (c) 84 (d) 98 (e) 46 (f) 32
16. Evaluate the following:

{
3
3
3
1
2
2
(a) { (5 + 12 2 ) } (b) ( 10 - ) (c) (10 + 24 2 ) } (d) (37 - 35 2 ) 3 2
2
1
6
3
2
3
2
17. Write the units digit of the cube of each of the following numbers:
(a) 33 (b) 132 (c) 995 (d) 654 (e) 1,999 (f) 2,008
Cubes of Negative Integers
3
The cube of a negative integer is always negative, i.e., (–a) = (–a) × (–a) × (–a) = –a 3
3
3
For example, (i) (–4) = (–4) × (–4) × (–4) = –64 (ii) (–5) = (–5) × (–5) × (–5) = –125
Example 6: Show that –1,66,375 is a perfect cube. Also find the number whose cube is –1,66,375.
Solution: Writing 1,66,375 as a product of its prime factors, we get 5 1,66,375
1,66,375 = 5 × 5 × 5 × 11 × 11 × 11
5 33,275
Grouping them into groups of three, you can see that no number is 5 6,655
left ungrouped.
So, 1,66,375 is a perfect cube of 5 × 11 = 55. 11 1,331
Also, (–55) × (–55) × (–55) = –1,66,375 11 121
3
\ (–55) = –1,66,375 11 11
Hence, –1,66,375 is a perfect cube and it is a cube of –55. 1
Cubes of Rational Numbers

p Ê pˆ 3 p p p p ¥ p ¥ p p 3
If q is a rational number, q ≠ 0, then Á ˜ = q ¥ q ¥ q = qqq = q 3 .
¥¥
Ë
q ¯
¥¥
Ê 4ˆ 3 444 4 () 3 64
For example, Á ˜ = 555 = 5 () 3 = 125
5¯
Ë
¥¥
-p
If q is a negative rational number, q ≠ 0, then
Ê - ˆ p 3 Ê - ˆ p Ê - ˆ p Ê - ˆ p ( - p) ¥ - p( ) ¥ - p( ) ( - p) 3 - p 3
Á q ˜ ¯ = Á q ˜ ¯ ¥ Á q ˜ ¥ Á q ˜ ¯ = q ¥ q ¥¥ q = q 3 = q 3
¯
Ë
Ë
Ë
Ë

(
(
Ê - ˆ 2 3 ( -2) ¥ -2) ¥ -2) -8
For example, Á Ë 3 ˜ = 3 ¥¥ 3 = 27
¯
3
69