Sqaure And Square Roots, Cube And Cube Roots

Sqaure And Square Roots, Cube And Cube Roots

Cube roots
9. If ‘n’ is perfect cube then for some integer ‘n’. n = m³, then the number m is called the cube root of ‘n’ such that \( n = m^3 \Rightarrow m = \sqrt[3]{n} \)     
    Ex : 3 is a cube root of  27\( \Rightarrow \sqrt[3]{{27}} = \sqrt[3]{{3^3 }} \) 
                =\( \left( {3^3 } \right)^{\frac{1} {3}} = 3 \)
*  Find the cube root of a two digit number 
    If ab is a number alternate method i.e a is the tens digit of the number and ‘b’ is the units digit. We find cube by using.
    (a+b)³ = a³ +3a²b +3ab²+b³ for (ab)³ we make k’4’ columns
Ex :   Find the value of (23)³

\( \therefore \)required number = 12, 167
\( \sqrt[3]{{12,167}} = 23 \)

Example 2 : Find 98³ by using altemate 
Sol :  a = 9, b = 8
(a+b)³ = a³ + 3a²b +3ab² +b³


  941                   1                   9                 2
\( \therefore (98)^3 = 941192 \Rightarrow \sqrt[3]{{9411192}} = 98 \)

* Method of sucessive subtraction to find cube root :
    We subtract the numbers, 1, 7,19, 37, 61, 91, 129, 169..... sucessvily till we get zero. Then the number of subtrections will give the cube roots of the number. These numbers are given by putting n = 1,2,3.... in 1+n(n-1)+3
Ex : Find the cube root of 343 by using the method of successive subtraction.
Sol : 1) 343 - 1 = 342,        2) 342 - 7 = 335,     3) 335- 19 = 316
    4) 316 - 37 = 279,    5) 279 -- 61 = 218,    6) 218 = 91 = 127
    7) 127- 127= 0        
* Cube root by prime factorisation method.
    Let ‘n’ be a number its cube root 
    1) Find the prime factors of n
    2) group the factors in triplets such that all the three factors intriplet are the same
    3) If some prime factors are left ungrouped, then the number ‘n’ is not a perfect cube and the process stops.
    4) If no, factor is left ungrouped then choose one factor from each group and take the product then product is cube root of ‘n’
Ex : 27, 000
    = 3 x 3 x 3 x 2 x 2x 2x 5 x 5 x 5               \( \therefore \sqrt[3]{{27000}} = 3 \times 2 \times 5 = 30 \)