Square And Square Roots, Cube And Cube Roots

Square And Square Roots, Cube And Cube Roots

Main Points Related to  Squares, Square roots and cube, cube roots
1.  A number is multiplied by it self the product so obtained is called the square of that number
    Ex: \( 1)\,\,\,\,a\,\, \times \,\,a\,\,\, = a^2 \,\,\,(a \in R) \)        2)\( \sqrt 2 \sqrt 2 = \left( {\sqrt 2 } \right)^2 = 2 \)
2. A natural number is called  a perfect square, if it is the square of some naratural 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)    =\( 2.\frac{{n(n + 1)}} {2} = n(n + 1) \)
8.  A number having 2,3,7 (or) 8 at its units place it never be a perfect square 
    Ex :257, 368 etc
9.  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\( \Rightarrow q = \sqrt[3]{a} \) 
10. Cube of even natural number is even
      Ex : 4 x 4 x 4 = 4³ = 64
11. Cube of negative number is negative 
      Ex: \( - \frac{7} {5} \times - \frac{7} {5} \times - \frac{7} {5} = - \frac{{343}} {{125}} \)
12.  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)\( \sqrt[3]{{ - 1}} = - 1\,\,\,\,\,and\,\,\,\sqrt[3]{1} = 1 \)
13. The cube root of a negative perfect cube is negative
      Ex : \( \sqrt[3]{{ - 125}} = \sqrt[3]{{( - 5)^3 }} = - 5 \)
14. 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
15. 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
16. 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
17. The cube of a number ends in zero will end in zeros
      Ex : 10³=  1000,     30³ = 9000
18. If ‘a’ is a perfect cube and for some integer ‘b’ a = b³ , then the number ‘b’ is called cube roots of \( a \Rightarrow b = \sqrt[3]{a}\,\,(or)\,\,\,a^{\frac{1} {3}} \)