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SURDS AND INDICES PROBLEMS

Aptitude

SURDS AND INDICES -> IMPORTANT FACTS AND FORMULAE

1. LAWS OF INDICES :
  (i) am * an = am + n
  (ii)  am / an = am - n
  (iii)  (am)n = amn
  (iv)  (ab)n = anbn
  (v)  (a/b)n = an/ bn
  (vi)  a0 = 1
2. SURDS : Let a be rational number and n be a positive integer such
that a(1/n) = n√a
3 LAWS OF SURDS :
  (i)  n√a = a (1/n)
  (ii)  n√ab = n√a x n√b
  (iii)  n√a/b = n√a / n√b
  (iv)  (n√a)n = a

SURDS AND INDICES -> EXAMPLES

1. What is the quotient when (x-1 - 1) is divided by (x - 1)?
  Sol. x-1 - 1 / x-1 - (1/x)-1 / x-1 = (1-x)/x =(1-x)/x * 1/(x-1) = -1/x.
Hence, the required quotient is - 1/x
2. If 2x-1 + 2x+1 = 1280, then find the value of x
  Sol. 2x-1 + 2x+1 = 1280 ⇔ 2x-1(1+2x) = 1280
⇔ 2x-1 = 1280/5 = 256 = 28 ⇔ x-1 = 8 ⇔ x = 9.
Hence, x = 9.
3. If [1/5]3y = 0.008, then find the value of (0.25)y
  Sol. [1/5]3y = 0.008 = 8/1000 = 1/125 = [1/5]3 ⇔ 3y = 3 ⇔ y = 1.
∴ (0.25)y = (0.25)1 = 0.25.
Hence, the required distance is 6 km.

SURDS AND INDICES -> EXERCISE

1. The value of (256)5/4 is:
 
  • A. 11 hrs 562 min
  • B. 3 hrs 860 min
  • C. 7 hrs 1020 min
  • D. 7 hrs 1024 min
Ans: D.
Sol.
(256)5/4 = (44)5/4 = 4(4*5/4) = 45 = 1024.
 
2. If 5a= 3125, then the value of 5(a-3) is
 
  • A. 25
  • B. 50
  • C. 250
  • D. 300
Ans: A.
Sol.
5a = 3125 = 5a = 55
a = 5.
5(1-3) = 5(5-3) = 52 = 25.
 
 
3. The value of 51/4 * (125)0.25 is:
 
  • A. 5
  • B. 55
  • C. 5√5
  • D. √5
Ans: A.
Sol.
51/4 * (125)0.25 = 50.25 * (53)0.25 * 5(3*0.25) = 50.25 * 50.75
= 5(0.25 + 0.75) = 51=5.