CBSE - Class 8 - Maths - Cubes and Cube Roots

Important Points 
& NCERT Solutions

1. If n is a perfect cube then n = m³ or m is the cube root of n. i.e. (n = m × m × m) 

2. A cube root is written as ∛n or n^1/3.

3. ∛2, ∛3, ∛4 etc. all are irrational numbers.

4. The cube root of negative perfect cube is negative. i.e.

    (-x)³=  -x³ 

5. Square root of a negative number is not a real number but cube root of a negative number is a real number.

6. ∛(ab) = ∛a × ∛b

7.  Even number has even cube while odd number has odd cube number. There are only ten perfect cubes from 1 to 1000.

Number      n	Perfect Cube         n3	Number        n	Perfect Cube n3
1	1	11	1331
2	8	12	1728
3	27	13	2197
4	64	14	2744
5	125	15	3375
6	216	16	4096
7	343	17	4913
8	512	18	5832
9	729	19	6859
10	1000	20	8000

8. Each prime factor appears three times in its cubes

e.g.

    4³ = 64 = 2 × 2 × 2 × 2 × 2 × 2 = 2³ × 2³

2744 = 2 × 2 × 2 × 7 × 7 × 7 = 2³ × 7³

9. The sum of any number of consecutive cubes, beginning with 1, is always a square number.
e.g.
1 + 8 = 9 = 3²

1 + 2³+ 3³= 1 + 8 + 27 = 36 = 6²
1 + 2³+ 3³ + 4³= 1 + 8 + 27 + 64= 100 = 10²

1 + 2³+ 3³ + ... + n³= [n(n+1)/2]²

10. Ramanujan number: 1729
Once mathematician Prof. G.H. Hardy came to visit him in a taxi whose taxi number was 1729. While talking to Ramanujan, Hardy described that the number 1729 was a dull number. Ramanujan quickly pointed out that 1729 was indeed an interesting number. He said, it is the smallest number that can be expressed as a sum of two cubes in two different ways.
i.e.  1729 = 1728 +1 = 12³ + 1³
and 1729 = 1000 + 729 = 10^{3 +}  9³




Q1 (NCERT): State true or false.
(i) Cube of any odd number is even.
(ii) A perfect cube does not end with two zeros.
(iii) If square of a number ends with 5, then its cube ends with 25.
(iv) There is no perfect cube which ends with 8.
(v) The cube of a two digit number may be a three digit number.
(vi) The cube of a two digit number may have seven or more digits.
(vii) The cube of a single digit number may be a single digit number.

Answer:

(i) False.

(ii) True

(iii) False

(iv) False (e.g. 15³ = 3375, 15² = 225)

(v) False

(vi) False ( 99³ = 970299 )

(vii) True (1³ = 1)

Q2: Is 1080 a perfect cube? If not, with what number it should be multiplied or divided to make it a perfect cube?

Answer: Prime factors of 1080 are:

1080 = 2 × 2 × 2 × 3 × 3 × 3 × 5 = 2³ × 3³ × 5

Since one factor is left, it is not a perfect cube.

Either we divide the number by 5 to make it perfect cube or multiply by 25

i.e.

1080 ÷ 5 = 216 = 2³ × 3³ = (2 × 3)³ = 6³

1080  × 25 = 2³ × 3³ × 5³ = (2 ×3 × 5)³ = (30)³

Q3: What is the smallest number by which 392 must be multiplied so that the product is a perfect cube?

Answer: Prime factors of  392 = 2 × 2 × 2 × 7 × 7 

To make it a perfect cube, it must be multiplied by 7.

Q4: Find the cube root of 0.027

Answer: ∛(0.027) 

   = ∛(27/1000) 

   = ∛(27) / ∛(1000) 

   = ∛(3 × 3 × 3) / ∛(10 × 10 × 10) 

   = 3/10 

   = 0.3

11. Instant Cube Root (for 6-digit numbers)

1. Consider a 6 digit perfect cube number.  (e.g. 262144)
2. Divide the number into two groups (each of three digit).  (262 and 144)
3. Check the unit digit of the first group (from right) and find the cube associated with unit digit. i.e. 144 the unit digit is 4. The cube of 4 is 4.
4. Check the second group. 262 we know 6³= 216 and 7³ = 343. It means, 6³ <262 < 7³. Consider the  smaller number, i.e.  6 for tens place.
5. We can guess the cube of 262144 is 64³

Q5: Three numbers are in the ratio 1:2:3. The sum of their cubes is 98784. Find the numbers.

Answer: Let the numbers be x, 2x and 3x.

Sum of their cubes is: x³ + (2x)³ + (3x)³  = 987784

⇒ x³ + 8x³ + 27x³  = 987784

 36x³  = 987784

 x³  = 987784 / 36 = 2744

x = ∛2744 = 14

Famous Mathematician G. H. Hardy, wrote in his book 'A Mathematician's Apology',
There are just four numbers, after unity, which are the sums of the cubes of their digits:
153 = 1³ + 5³ + 3³
370 = 3³ + 7³ + 0³
371 = 3³ + 7³ + 1³
407 = 4³ + 0³ + 7³
These numbers are part of  Narcissistic numbers.