SURDS AND INDICES

Simple problems:


1.  Laws of Indices:
      
      (i)    am * an = a(m+n) 
      (ii)   am / an = a(m-n)
      (iii)  (am)n   = a(m*n)
      (iv)   (ab)n   = an * bn
      (v)    (a/b)n  = an / bn
      (vi)   a0      = 1


2.Surds :Let  'a' be a rational number & 'n' be a positive 
  integer such that a1/n = nth root a is irrational.Then nth 
  root a is called 'a' surd of 'n'.

    Problems:-
     (1)
     (i)  (27)2/3 = (33)2/3 = 32 = 9.
     (ii) (1024)-4/5 =  (45)-4/5 = (4)-4=   1/(4)4 = 1/256.
     (iii)(8/125)-4/3 =((2/5)3)-4/3 = (2/5)-4 = (5/2)4 = 625/16 

(2) If 2(x-1)+ 2(x+1) = 1280 then find the value of x .


   Solution: 2x/2+2x.2 = 1280
             2x(1+22) = 2*1280
             2x = 2560/5
             2x  =  512  => 2x = 29
             x = 9


(3)  Find the value of [5[81/3+271/3]3]1/4
        
   Solution: [5[(23)1/3+(33)1/3]3]1/4 
                   [5[2+3]3]1/4
                   [54]1/4  => 5.


(4) If (1/5)3y= 0.008 then find the value of (0.25)y
  
   Solution:  (1/5)3y  = 0.008 
       (1/5)3y =[0.2]3  
              (1/5)3y =(1/5)3  
               3y= 3   =>   y=1.  
              (0.25)y = (0.25)1 =>  0.25 = 25/100  = 1/4

      
(5) Find the value of (243)n/5 *  32n+1 /  9n  * 3 n-1
               
   Solution:   (35)n/5  *  32n +1  /  (32)n  * 3n-1  
                 33n+1 / 33n-1 3
                 33n+1 * 3-3n+1    => 32   =>9.


(6) Find the value of  (21/4-1)( 23/4 +21/2+21/4+1)
        
    Solution:  Let us say  21/4  = x   
              (x-1)(x3+x2+x+1)
              (x-1)(x2(x+1)+(x+1))
              (x-1) (x2+1) (x+1)   [(x-1)(x+1) = (x2-1)]
              (x2+1) (x2-1)  => (x4-1)  
              ((21/4))4 - 1)  = > (2-1)  = > 1.


(7)  If x= ya , y = zb  , z = xc  then find the value
     of abc.

    Solution:   z= xc    
                z= (ya)c [  x= ya ]
                z= (y)ac  
                z= (zb)ac   [y= zb]
                z= zabc   
                abc  = 1


(8)Simplify (xa/xb)a2+ab+b2*(xb/xc)b2+bc+c2*(xc/xa)c2+ca+a2

   Solution:[xa-b]a2+ab+b2 * [xb-c]b2+bc+c2 * [xc-a]c2+ca+a2 
       
       [ (a-b)(a2+ab+b2)  = a3-b3]  
             
   from the above formula 
           =>   xa3-b3   xb3-c3 xc3-a3
           => xa3-b3+b3-c3+c3-a3         
           =>   x0   =    1


(9) (1000)7   /1018    = ?

     (a) 10        (b)  100    (c ) 1000    (d) 10000 
           
   Solution: (1000)7 / 1018
             (103)7 /  (10)18  = >  (10)21 /  (10)18   
              
              =>  (10)21-18    => (10)3 =>  1000  
   Ans :( c )


(10) The  value  of   (8-25-8-26)  is
    
    (a) 7* 8-25  (b) 7*8-26  (c ) 8* 8-26  (d) None  

   Solution: ( 8-25 -  8-26  )
              =>  8-26 (8-1 )
              =>  7* 8-26 
   Ans: (b)
                                                          

(11)   1 / (1+ an-m ) +1/ (1+am-n)    = ?  
       (a)  0    (b)  1/2    (c ) 1   (d)    an+m 
                      
   Solution:  1/  (1+ an/am)  + 1/ (  1+ am/an)
               => am / (am+ an ) +  an  /(am +an )  
               => (am  +an )   /(am +  an) 
               => 1
   Ans: ( c) 


(12) 1/(1+xb-a+xc-a)+1/(1+xa-b+xc-b)+1/(1+xb-c+xa-c)=?
     
    (a) 0  (b) 1  ( c ) xa-b-c (d) None  of the above

   Solution:  1/ (1+xb/xa+xc/xa) + 1/(1+xa/xb +xc/xb) + 
              1/(1+xb/xc +xa/xc)
       => xa /(xa +xb+xc) + xb/(xa +xb+xc) +xc/(xa +xb+xc)
       =>(xa +xb+xc) /(xa +xb+xc)
       =>1
   Ans:  (b)


(13) If x=3+2 √2  then the value of (√x – 1/ √x)
     is [ √=root]

      (a)  1   (b)   2    (c )  2√2    ( d) 3√3

   
    Solution:  (√x-1/√x)2  =  x+ 1/x-2
             =>  3+2√2 + (1/3+2√2 )-2 
             =>  3+2√2  + 3-2√2 -2
             =>  6-2 = 4
                (√x-1/√x)2  = 4    
             =>(√x-1/√x)2  = 22 
                (√x-1/√x)  =  2.
    Ans : (b)


(14)  (xb/xc)b+c-a (xc/xa)c+a-b (xa/xb)b+a-c  = ?

       (a) xabc  (b) 1  ( c) xab+bc+ca (d) xa+b+c 

 
 Solution: [xb-c]b+c-a    [xc-a]c+a-b    [xa-b]a+b-c  
   
        =>x(b-c)(b+c-a)   x(c-a)(c+a-b)  x(a-b)(a+b-c) 
        =>x(b2-c2-ab-ac)    x(c2-a2-bc-ab)   x(a2-b2-ac-bc)
        =>x(b2-c2-ab-ac+c2-a2-bc-ab+a2-b2-ac-bc)
                             =>  x0  
                             =>1
     Ans: (b)
  
(15)  If 3x-y  = 27 and  3x+y  = 243 then x is equal to
       
       (a)   0  (b)  2    (c ) 4      (d) 6 

   Solution:  3x-y  = 27    =>    3x-y    =  33 
              x-y= 3
              3x+y   = 243 =>    3x+y  = 35 
              x+y = 5
              From above two equations  x = 4 , y=1
   Ans: (c )


(16) If ax = by = cz and  b2 = ac then  ‘y’equals
         
     (a)xz/x+z (b)xz/2(x-z) (c)xz/2(z-x) (d)2xz/x+z 

   Solution:  Let us say  ax = by  = cz = k
              ax  =k   =>  [ax]1/x = k1/x 
              =>  a =  k1/x
              Simillarly  b = k1/y  
              c = k1/z 
              b2  =  ac   
              [k1/y]2=k1/xk1/z 
              =>k2/y = k1/x+1/z
              => 2/y = 1/x+1/y
              =>y= 2xz/x+z
   Ans:  (d) 


(17)   ax = b,by = c ,cz = a then the value of
       xyz is is
     
     (a) 0   (b) 1   (c ) 1/abc   (d) abc  
              
  Solution:    ax = b  
       (cz)x  = b           [cz = a] 
               by)xz = b           [by = c] 
               =>xyz =1
  Ans: (b) 


(18) If 2x = 4y =8z and (1/2x +1/4y +1/6z) =24/7 then
     the value of 'z' is
        
     (a)  7/16   (b)  7 / 32   (c ) 7/48  (d) 7/64  
             
   Solution:  2x = 4y=8z
              2x = 22y = 23z  
              x= 2y = 3z
              Multiply above equation with ‘ 2’
              2x = 4y= 6z
              (1/2x+1/4y+1/6z) = 24/7
              =>(1/6z+1/6z+1/6z) = 24/7
              =>  3 / 6z = 24/7
              => z= 7/48
    Ans: ( c)