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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

7. If √2n = 64, then the value of n is :
 
  • A. 6
  • B. 12
  • C. 24
  • D. 48
Ans: B.
Sol.
√2n = 64 ⇔ (2n)1/2 ⇔ 2n/2 = 26 ⇔ n/2
= 6 ⇔ n = 12.
 
8. If 2n+4 - 2n+2 = 3, then n is equal to :
 
  • A. 2
  • B. -2
  • C. 1
  • D. -1
Ans: B.
Sol.
2n+4 - 2n+2 = 3 ⇔ 2n+2 (22 - 1) = 3 ⇔ 2n+2 = 1 = 20
⇔ n+2 = 0 ⇔ n= -2.
 
 
9. Given that 100.48 = x, 100.70 = y and xz = y², then the value of z is close to
 
  • A. 1.2
  • B. 1.45
  • C. 2.2
  • D. 2.9
Ans: D.
Sol.
xz = y² ⇔ (10 0.48 )z(100.70)2 (10 0.48z) = 10(2×0.70) = 101.40
⇔ 0.48z = 1.40 ⇔ z= 140/48 = 35/12 = 2.9 (approx).