Page 57 - ICSE Math 8

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 3: 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. 11 1,331
So, 1,66,375 is a perfect cube of 5 × 11 = 55. 11 121
Also, (–55) × (–55) × (–55) = –1,66,375 11 11
3
\ (–55) = –1,66,375 1
Hence, –1,66,375 is a perfect cube and it is a cube of –55.

Cubes of Rational Numbers
p Ê pˆ 3 p p p p ¥ p ¥ p p 3
If is a rational number, q ≠ 0, then = ¥ ¥ = = .
q Á ˜ q q q qqq q 3
¥¥
Ë
q ¯
¥¥
Ê 4ˆ 3 444 4 () 3 64 Try This
For example, Á ˜ = = 3 =
5¯
Ë
¥¥
−p 555 5 () 125 Find the cubes of:
1
If is a negative rational number, q ≠ 0, then (a) 2 (b) –9 (c) 5.2
q 3 8
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 3 = 27
¯
EXERCISE 4.1
1. Find the cubes of the following: (a) –12 (b) –23 (c) –35 (d) –42
2. Which of the following are cubes of negative integers?
(a) 343 (b) 1,331 (c) 4,096 (d) 344 (e) 24,000 (f) 588
3. Show that the following integers are cubes of negative integers. Also find the integers whose cubes they
represent.
(a) –6,859 (b) –97,336 (c) –21,952 (d) –17,71,561 (e) –8,57,375
4. Find the cubes of the following numbers.
1 1
(a) 6.3 (b) –0.21 (c) 3 (d) −5 (e) 0.08 (f) –7.02
4 6
5. Is 1,728 a perfect cube? Also find the number whose cube is 1,728.
Cube Roots

The cube root of a number is that number which when cubed gives the original number. In other words, if
3
3
n is the cube of a number n, then n is the cube root of n . Cube root of n is denoted as n. 3 is called the
3
radical. 3 is called the index of the radical. The number whose cube root we wish to find is called the radicand.
8
8 ˆ
2ˆ
3
For example, 27 = 3 fi 3 27 = 3, Ê Á Ë 343¯ = Ê Á ˜ 3 fi 3 343 = 2 and so on.
˜
7¯
Ë
7
index
radicand
45