Download Cube and Cube Roots

Cube and Cube Roots
Main Concepts and Results:
• A natural number is called a perfect cube if it is the cube of some natural number
i.e., if m =
, then m is a perfect cube where m and n are natural numbers.
• Number obtained when a number is multiplied by itself three times are called cube
number.
• A perfect cube can always be expressed as the product of triplets of prime factors.
• The cube root of a number x is the number whose cube is x. It is denoted by √ .
• Cubes of the numbers ending with the digits 0, 1, 4, 5, 6 and 9 end with digits 0, 1, 4, 5, 6
and 9 respectively.
• Cube of the number ending in 2 ends in 8 and cube root of the number ending in 8 ends
in 2.
• Cube of the number ending in 3 ends in 7 and cube root of the number ending in 7 ends
in 3.
Exercises:
1) What is Hardy – Ramanujan Number? What is interesting about this number?
2) What are perfect cubes or cube numbers?
3) Complete the below mentioned table:
4) Find the one’s digit of the cube of each of the following numbers.
(i)
3331
(ii)
8888
(iii)
149
(iv) 1005
(vi)
77
(vii) 5022
(viii) 53
(v)
1024
5) Express the following numbers as the sum of odd numbers adding consecutive odd
numbers.
(a) 6
(b) 7
(c) 8
6) Consider the following pattern.
2 –1 = 1 + 2 × 1 × 3
3 –2 = 1 + 3 × 2 × 3
4 –3 = 1 + 4 × 3 × 3 Using the above pattern, find the value of the following.
7) Is 243 a perfect cube?
8) Which of the following are perfect cubes?
9) Is 392 a perfect cube? If not, find the smallest natural number by which 392 must be
multiplied so that the product is a perfect cube.
10) Is 53240 a perfect cube? If not, then by which smallest natural number should 53240 be
divided so that the quotient is a perfect cube?
11) Is 1188 a perfect cube? If not, by which smallest natural number should 1188 be divided so
that the quotient is a perfect cube?
12) Is 68600 a perfect cube? If not, find the smallest number by which 68600 must be multiplied
to get a perfect cube.
13) Is 9720 a perfect cube? If not, find the smallest number by which it should be divided to get
a perfect cube.
14) Write two Pythagorean triplets each having one of the numbers as 5.
15) By what smallest number should 3600 be multiplied so that the quotient is a perfect cube.
Also find the cube root of the quotient.
16) If one side of a cube is 15m in length, find its volume.
17) Find the length of each side of a cube if its volume is 512
.
18) Three numbers are in the ratio 1:2:3 and the sum of their cubes is 4500. Find the numbers.
19) Three numbers are in the ratio 2:3:4. The sum of their cubes is 0.334125. Find the numbers.
20) Difference of two perfect cubes is 189. If the cube root of the smaller of the two numbers is
3, find the cube root of the larger number.
21) Check which of the following are perfect cubes. (i) 2700 (ii) 16000 (iii) 64000 (iv) 900 (v)
125000 (vi) 36000 (vii) 21600 (viii) 10,000 (ix) 27000000 (x) 1000. What pattern do you
observe in these perfect cubes?
22) Which of the following numbers are not perfect cubes?
(i)
216
(ii) 128
(iii) 1000
(iv) 100
(v) 46656
23) Find the smallest number by which each of the following numbers must be multiplied to
obtain a perfect cube.
(i)
243
(ii) 256
(iii) 72
(iv) 675
(v) 100
24) Find the smallest number by which each of the following numbers must be divided to obtain
a perfect cube.
(i)
81
(ii) 128
(iii) 135
(iv) 192
(v) 704
25) Parikshit makes a cuboid of plasticine of sides 5 cm, 2 cm, 5 cm. How many such cuboids will
he need to form a cube?
26) What are cube roots? Also mention its symbol.
27) Find the cube root of 8000 and 74088.
28) Find the cube root of 13824 by prime factorisation method.
29) Find the cube root of 17576 through estimation.
30) Find the cube root of each of the following numbers by prime factorisation method.
31) 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.
32) You are told that 1,331 is a perfect cube. Can you guess without factorisation what is its
cube root? Similarly, guess the cube roots of 4913, 12167, 32768.
33) Evaluate :
34) Evalutae:
(i)
(ii)
35) Using prime factorisation, find which of the following are perfect cubes.
(i)
128
(b) 343
(c) 729
(d) 1331
36) Write the Pythagorean triplet whose one of the numbers is 4.
37) Using prime factorisation, find the cube roots of
(i)
512
(b) 2197
38) State whether the statements are true (T) or false (F).
(I)
The cube of 0.4 is 0.064.
(II)
The cube root of 8000 is 200.
(III)
There are five perfect cubes between 1 and 100.
(IV)
For every natural number m, (2m–1, 2 –2m, 2 –2m + 1) is a Pythagorean triplet.
(V)
All numbers of a Pythagorean triplet are odd.
(VI)
If x and y are integers such that > , then > .
(VII)
Let x and y be natural numbers. If x divides y, then
divides .
(VIII) A number having 7 at its ones place will have 3 at the ones place of its cube.
(IX)
The cube of a one digit number cannot be a two digit number.
(X)
Cube of an even number is odd.
(XI)
Cube of an odd number is even.
(XII)
Cube of an even number is even.
(XIII) Cube of an odd number is odd.
(XIV) 999 is a perfect cube.
(XV)
363 × 81 is a perfect cube.
(XVI) Cube roots of 8 are +2 and –2.
(XVII) There is no cube root of a negative integer.
(XVIII) Square of a number is positive, so the cube of that number will also be positive.
39) Fill in the blanks to make the statements true.
(I)
There are _________ perfect cubes between 1 and 1000.
(II)
The cube of 100 will have _________ zeroes.
(III)
1
= _________
.
(IV)
Ones digit in the cube of 38 is _________.
(V)
(1.2) = _________.
(VI)
The cube of an odd number is always an _________ number.
(VII)
The cube root of a number x is denoted by _________.
(VIII) . The least number by which 72 be multiplied to make it a perfect cube is
_____________.
(IX)
The least number by which 72 be divided to make it a perfect cube is
_____________.
(X)
Cube of a number ending in 7 will end in the digit _______________.
Multiple Choice Questions:
40) The one’s digit of the cube of 23 is
(a) 6
(b) 7
(c) 3
(d) 9
41) If one member of a pythagorean triplet is 2m, then the other two members are
42) Which of the following numbers is a perfect cube?
(a) 243
(b) 216
(c) 392
43) Which of the following numbers is not a perfect cube?
(a) 216
(b) 567
(c) 125
44) √1000 is equal to
(a) 10
(b) 100
45) If m is the cube root of n, then n is
46) The value of 45 - 65 - 20 is ___
(a) 175500
(b) -191500
(c) 1
(c) 170000
(d) 8640
(d) 343
(d) None of these
(d) -170000
47) Three number are in the ratio 2:3:4 to one another. The sum of their cubes is 33957.
48) The cube root of 27 is____
(a) 27
(b) 9
(c) 3
(d) None of the above
49) The cube of a number x is nine times of x, then find x, where x≠ 0 and x≠ -3.
(a) 8
(b) 2
(c) 4
(d) 3
50) If 36 =
, then is equal to,
(a)
(b) 6
(c)
(d) 6
51) Two cubes have volume in the ratio 1 : 27. The ratio of the area of the face of one to that of
the other is ____
(a) 1 : 3
(b) 1 :6
(c) 1 : 9
(d) 1 : 18
52) The volume of a cube whose surface area is 150
, is___
(a) 25
(b) 75
(c) 125
(d) 27
53) If
4
(a) 1
= x, then the value of x is ____
54) The value of
(a) 4.5
(
)
(b) 1
is ____
(b) 5.5
(c) 2
(c) 6.5
(d) 1
(d) 7.5