The document discusses cube numbers, their properties, cube roots and related concepts. It provides the following key points: 1) A cube number is a number multiplied by itself 3 times (m3). The cube of a positive number is positive, and the cube of a negative number is negative. 2) To find the cube root of a number is to find the number that when multiplied by itself 3 times equals the given number. 3) Finding the cube root of a number can be done by prime factorizing the number and taking the cube root of each prime factor. 4) The units digit of cube numbers follows a repeating pattern related to the original number, as shown in tables in the document

1. CUBE AND CUBE ROOTS
By Somu Rajesh

2. CUBE NUMBERS
A number multiplied by itself 3 times is known as acube number and is represented as m3
Eg.33 = 3× 3 × 3 = 27
Cube of apositive number is always positive
Cube of anegative number is always negativeeg.(-5)3 = (-5)(-5)(-5) = -125
A number is said to be aperfect cube ifit has factor triplets
Eg.216 canbe split as 23 × 33 ( factor triplets of 2 and 3) thus it is the cube of 2 × 3 = 6
Cubes of odd numbers are odd and cubes of evennumbers are even
(m cube)

3. UNITS DIGIT OF CUBENUMBERS
Numbers Cubes Units
plac
e
1 1 1
2 8 8
3 27 7
4 64 4
5 125 5
6 216 6
7 343 3
8 512 2
9 729 9
10 1000 0
Numbers Cubes Units
plac
e
11 1331 1
12 1728 8
13 2197 7
14 2744 4
15 3375 5
16 4096 6
17 4913 3
18 5832 2
19 6859 9
20 8000 0

4. PATTERNS RELATED TO CUBE NUMBERS
Sum of consecutive odd numbers
13= 1=1
23= 8 = 3+5
33 = 27 = 7+9+11
43 = 64 = 13+15+17+19
53 = 125 = 21+23+25+27+29
formula N*[n-1]+1

5. CUBE ROOTS
The cube root of anumber is the number which multiplied by itself 3 times gives the given number.
Cube root is the inverse of cube ie.ifx3 = y then 3 𝑦(cube root of y) = x
43 = 64 then 3
64 = 4
Geometrically cube root canbe visualised as the side of acube whose volume is known
Cube root of anegative number is negative of the cube root of its positive
i.e.(-m) (-m) (-m) = -m3 thus 3
−m3 = -m = - (cube root ofm3)

6. CUBE ROOT BY PRIMEFACTORIZATION
Step 1 – prime factorize thenumber 5
Step 2 – Form groups in triplet of identical primes 5
Step 3 – Takeone prime number from eachtriplet 5
Step 4 – Find the product of the numbers taken in step 3 3
3
3
Example – Find the cube root of 3375
3375 = 5× 5 × 5 × 3 × 3 × 3
Cube root of 3375 = 3 × 5 = 15
3375
675
135
27
9
3
1

7. CUBE ROOT OF FRACTIONS AND DECIMALS
Cube root of afraction number can be found by working out the cube root of the numerator and denominator
separately
Eg.Cube root of
1331 1331
= 3 =
343 3
343 7
1 1
Cube root of adecimalnumber can be found by converting the decimalto afraction and then findingthe cube
root of numerator and denominator separately
3
Example Cube root of .000027 = 27/1000000 = 3
1000000
3
27 3
100
= = 0.03

8. PRACTICE QUESTIONS
Find the cube root of
a) 4096 (b) 9261
Find the cube root of
a) -5832 (b) – 2197 (c) -2744000 (d) -512/343 (e) 729/-1331
Find the cube root of
a) 729 × 8000 (b) 3375 × 216 (c) -216 × -1728
Find the cube root of
a) -15.625 (b) 0.000002744 (c) -175.616
Find the smallest number by which 26244 should be divided to make it aperfect cube?
Find the least number by which 30375 must be multipliedto get aperfect cube?

9. EXTRA PRACTICE
7.Write true (T) or false (F) for the following statements:
(i) 392 is aperfectcube.
(ii)8640 is not aperfectcube.
(iii)No cube can end with exactly two zeros.
(iv)There is no perfectcube which ends in 4.
(v) For an integer a,a3is always greater than a2.
(vi) Ifaand b are integers such that a2>b2, then a3>b3.
(vii) Ifadivides b,then a3dividesb3.
(viii) Ifa2ends in 9,then a3ends in 7.
(ix) Ifa2ends in an even number of zeros, then a3ends in 25.
(x) Ifa2ends in an even number of zeros, then a3ends in an odd number of zeros.

10. EXTRA PRACTICE
1.What is the smallest number by which the following numbers must be multiplied,so that the products are perfect
cubes?
(i) 107811 (ii) 35721
2.What happens to the cube of anumber ifthe number is multipliedby
(i) 3 (ii) 4
Write the units digitof the cube of each of the following numbers:
31,109,388,4276,5922,77774,44447,125125125
4.Find the cube root of each of the following natural numbers
i)134217728 (ii) 48228544 (iii) 74088000 (iv) 157464
(v) 1157625 (vi) 33698267
5.Multiply 210125 by the smallest number so that the product is aperfect cube.Also, find out the cube root of the
product.
6.What is the smallest number by which 8192 must be divided so that quotient is aperfect cube?Also, findthe cube
root of the quotient so obtained.

11. PRACTICE QUESTIONS
Find the least number by which 1323 must be multiplied so that the product is aperfect cube?
Find the smallest number by which 8768 must be divided so that the quotient is aperfect cube?
Find the smallest number by which 27783 be multiplied to get aperfect cube
Find the least number by which 8640 must be divided so that the quotient is aperfect cube?
What is the smallest number by which 77175 must be multiplied so that the product is aperfect cube?
Parikshit makes acuboid of plasticine of sides 5cm, 2cm and 5 cm.How many such cuboids will he need to form a
cube?

12. HOMEWORK QUESTIONS
Find the cube root of
a) 512 b) 91125 c)110592 d) 46656
e) 175616 f) 13824 g) 27000 i) 91125
Guess the cube root of the following without factorizing
a) 1331 b) 12167 c) 4913
Find the cube root of
d) 32768
a) 343
512
Find the cube root of
a) 9.261
b) 3375x512 c) 729x8000 d) −5832
1331
b) -0.125 c) 0.000027 d) 250.047
Find the smallest number by which 26244 should be divided tso that the quotient is aperfect cube?

13. PRACTICE QUESTIONS
Tell weather the numbers are cube numbers (a)12(b) 42 (c) 54
(d)31
Find the cube of (a) 2.5 (b)1.1 (c) 0.03 (d) 0.4
Find the cube of (a) 1/6(b) 2 1/3(c) 4/11
Find the cubes of (a) -7 (b) -30 (c) -18 (d) - 21
Find which of the following is aperfectcube (a) 243 (b) 1728 (c) 24000 (d) 1938
Which of the following are cubes of an even number
216,729,3375,8000,125,343,4096 and 9261