Tuesday, July 17, 2012

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 n1/3.

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
11111331
28121728
327132197
464142744
5125153375
6216164096
7343174913
8512185832
9729196859
101000208000


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 = 103 +  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.


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