Square And Square roots, cube And cube roots

SQUARES AND SQUARE ROOTS
What is special about the numbers 4, 9, 25, 64 and other such numbers?
Since, 4 can be expressed as 2 × 2 = 2², 9 can be expressed as 3 × 3 = 3², all
 such numbers can be expressed as the product of the number with itself.Such numbers like 1, 4, 9, 16, 25, ... are known as square numbers

Palindrome : A palindrome is a word; phrase, a sentence or numerical that reads the same forward or backward.
Ex : NOON ,MALAYALAM,MADAM,1565
Square Root : It is the inverse operation of square.
let the number be n, then the square root of n is that number which when multiplied by itself gives n as the product.
Ex :  \(\sqrt{4}=2\)

Main Points Related to  Squares, Square roots
1.    A number is multiplied by it self the product so obtained is called the square of that number
   Ex : \(\begin{align} & 1)a\times a={{a}^{2}}\left( a\in R \right) \\ & 2)\sqrt{2}\sqrt{2}=\left( {{\sqrt{2}}^{2}} \right)=2 \\ \end{align} \)
2.    A natural number is called  a perfect square, if it is the square of some natural number
    Ex : 1) 144 = 12 x 12 = 12²         2)\(\sqrt{\frac{169}{289}}=\sqrt{\frac{{{13}^{2}}}{{{17}^{2}}}}=\sqrt{\frac{13}{17}}\times \sqrt{\frac{13}{17}}\) 
3.    A perfect square is never negative
    Ex : 441 = - 21 x -21 = 21 x 21 = 21²
4.    Square of even number is even 
    Ex : 202 = 400 (even number)
5.    Square of odd number is odd
    Ex : 11² = 11 x 11 = 121 (odd)
6.    Sum of first ‘n’ odd number s is n² 
   Ex : 1= 1² , 1+3 = 4 = 2²,    1+3+5= 9 = 3² 
    1+3+5+....(2n-1) = n2 
7.    Sum of first n even natural numbers is n(n+1)
    Ex :  2+4+6+....+2n
= 2(1+2+......+n)    =\(\frac{2(n+1)}{2}=n\left( n+1 \right)\)
8.    A number having 2,3,7 (or) 8 at its units place it never be a perfect square 
Ex : 257, 368 etc
                                                                Methods for finding square roots 
Method of successtive subraction:  We subtract the numbers 1,3,5,7..... successively till we get zero the number of subtrctions will give the square root of the number.
Ex : Find square root of 64
Sol :      64- 1 = 63.  63 -  3 = 60, 60-5 = 55, 55-7, 48, 48-9=39, 39-11=28, 28- 13 = 15, 15-15 = 0
     The number of subtractions to yeild zero is 8
                                      \(\sqrt{64}=8\) 
Prime factorisation method for finding square root
Take the number ‘(n)’ whose square root is required 
    1) write all the prime factors of n
    2) pair the factors such that primes in each pair this formed is equal
    3) Choose one prime from each pair and multiply all such primes 
    4) The product of these primes is the square root of ‘n’
    Ex : Find S.R of  2025
    2025 = 5 x 5 x 3 x 3 x 3 x 3 = 5² x (32)² 
    \(\sqrt{2025}=5 \times9=45\)

Division method for finding square root
    If the given number has more than five digits, it is difficult to obtain prime factors, to over come this difficulty we use an alternate method is called division method 
    Procedure :
1.    Palce a bar over every pair of digits starting from the units digit
2.    Find the largest number whose square is less than or equal to the number under the bar to the extreme left
3.    Take this number as the division as well the quotient and the number under the bar to the extreme left as the divide and get the remainder
4.    Bring down the number under the next bar to the right of the remainder and this is the new divided
5.    Double the quotient and enter it with a blank on the right for the next deigit of the next possible divisor
6.    Guess the largest possible digit to fill the blank and also to become the new digit in the quotient, now we get a remainder.
7.    Bring down lthe number under the next bar to the right of the new remainder 
    Ex : Find the square root of 271441


Square Root of a decimal number (perfect square)
Procedure :
    1) Place bars on the integral part of the number in the usual manner 
    2) Place bars on the decimal part on every pair of digits beginning with the first decimal place
    3) find the square root by the division method as usual.
Ex : a) Find the square root of 1.5129        b) Find the square root of 0.000484


CUBE AND CUBE ROOTS
Introduction :
    This is a story about one of India’s great mathematical geniuses, S. Ramanujan.
    Once another famous mathematician Prof. G.H. Hardy came to visit him in a taxi whose number was 1729. While talking to Ramanujan, Hardy described this number “a dull number”. Ramanujan quickly pointed out that 1729 was indeed interesting. He said it is the smallest number that can be expressed as a sum of two cubes in two different ways:
    1729 = 1728 + 1 = 12³ + 1³       and     1729 = 1000 + 729 = 10³ + 9³
Cubes : A natural number ‘a’ is called a perfect cubes if there exists another number ‘b’ such that a = b X b X b = b³.In a simple language, if we multiply a number by itself three times , we get the cube of a number
Cube roots
    Cubes : A natural number ‘a’ is called a perfect cubes if there exists another  ‘b’ such that a = b X b X b = b³. In a simple language, if we multiply a number by itself three times , we get the cube of a number
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  \(n={{m}^{3}}\Rightarrow m=3\sqrt{n}\)

Main Points Related to  Cubes, Cube roots
1.    A number nultiplied by it self three times the product so obtained is called cube of that number
    Ex : n = a x a x a = a³ \(\Rightarrow q=3\sqrt{a}\)
2.    Cube of even natural number is even
    Ex : 4 x 4 x 4 = 4³ = 64
3.    Cube of negative number is negative 
\(-\frac{7}{5}\times -\frac{7}{5}\times -\frac{7}{5}=-\frac{343}{125}\)
4.    If a,b \(\in z\) then 
    a) (ab)³ = a³  x b³         b)  \({{\left( \frac{a}{b} \right)}^{3}}=\frac{{{a}^{3}}}{{{b}^{3}}}\)      c)\(\sqrt[n]{ab}=\sqrt[n]{a}\times \sqrt[n]{b}\)
    d)\(\sqrt[n]{\frac{a}{b}}=\frac{\sqrt[n]{a}}{\sqrt[n]{b}}\,\,\,\left( b\ne 0 \right)\)        e) \(\sqrt[m]{\sqrt[n]{a}}=\sqrt[mn]{a}\)       f)\(\sqrt[3]{-1}=-1\,\,\,\,\,and\,\,\,\sqrt[3]{1}=1\)
5.    The cube root of a negative perfect cube is negative
    Ex : \(\sqrt[3]{-125}=\sqrt[3]{{{(-5)}^{3}}}=-5\)
6.    The cubes of the digits 1,4,5,6 and 9 in units place are the 1 number ending in the same digits
    Ex : 1³ = 1, 4³ = 64, 5³ = 125, 6³ =216, ........9³ = 729
7.    The cubes of 2 in units place ends in 8 and the cube of 8 in units places ends in 2 
    Ex : 2³ = 8, 8³ = 512
8.    The cube of 3 in units place ends ‘7’ and the cube of 7 in units place ends in 3,
    Ex : 3³ = 27, 7³ = 343
9.    The cube of a number ends in zero will end in zeros
    Ex : 10³=  1000,     30³ = 9000
10.     If ‘a’ is a perfect cube and for some integer ‘b’ a = b³ , then the number ‘b’ is called cube root of a 
\(\Rightarrow b=\sqrt[3]{a}\,\,(or)\,\,\,{{a}^{\frac{1}{3}}}\)
* 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 \(\therefore \sqrt[3]{27000}=3\times 2\times 5=30\)

1. How to find cube root of a perfect cube number?
Prime factorization method :
i) For smaller numbers 
Ex:\(​​\sqrt[3]{216}\)
    216 = 2 x 2 x 2 x 3 x 3 x 3 = 2³  x 3³ = (2 x 3)³ = 6³

^{\(\therefore \sqrt[3]{216}=6\)}\(\begin{align} & 2\left| \!{\nderline {\, 216 \,}} \right. \\ & 2\left| \!{\nderline {\, 108 \,}} \right. \\ & 2\left| \!{\nderline {\, 54 \,}} \right. \\ & 3\left| \!{\nderline {\, 27 \,}} \right. \\ & 3\left| \!{\nderline {\, 9 \,}} \right. \\ & \text{ }3 \\ \end{align} \)
Estimation method
ii) For larger numbers : 

Ex : Find the cube root of 857375.

Solution: We will divide the number 857375 into two groups, i.e., 857 & 375.

375 will give a unit place digit of the required cube root and 857 will give the ten’s place digit.

Now 375 has 5 at the unit place. We know the cube of 5 will give the cube root as 5. So at the unit place, we get 5 for the required cube root.

 Now we take another group which is 857. So, from the cubes table, we observe the value 857 lies between cube of 9 and 10, i.e., 729 < 857 < 1000. So, we take here a lower number which is 9
Hence,  \(\sqrt[3]{\text{857375}}\)= 95
Cube Root of Non-Perfect Cubes
    There are many numbers which are not perfect cubes and we cannot find the cube root of such numbers using the prime factorisation and estimation method. Hence, we will use here some special tricks to find the cube root
Let us find the cube root of 150 here. Clearly, 150 is not a perfect cube
Step 1:  Now we would see 150 lies between 125 (cube of 5) and 216 (cube of 6). So, we will consider the lower number here, i.e. 5
Step 2:  Divide 150 by square of 5, i.e., 150/25 = 6
Step 3:  Now subtract 6 from 5 (whichever is greater) and divide it by 3. So, 6-5 = 1 &  \(1 \over 3\)= 0.333
Step 4:   At the final step, we have to add the lower number which we got at the first step and the decimal number obtained
So, 5+0.33 = 5.3
Therefore, the cube root of 150 is \(\sqrt[3]{\text{150 }}\) = 5.3
Find the cube root of the following decimal numbers:
Hint: Here we need to find the cube root of the given decimal number. For that, we will first write the decimal number as a fraction with numerator and denominator. Then we will find the cube root of the numerator and denominator using the factorization method. After getting the cube root of the numerator and denominator, we will again write the fraction as a decimal number.
Ex : 0.003375 = \(\frac{3375}{1000000}=\frac{5\times 5\times 5\times 3\times 3\times 3}{10\times 10\times 10\times 10\times 10\times 10}\)
\(\sqrt[3]{0.003375}=\frac{5\times 3}{10\times 10}=\frac{15}{100}=0.15\)