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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

6. (1000)7 ÷ 1018 = ?
 
  • A. 100
  • B. 1000
  • C. 10000
  • D. 100000
Ans: B.
Sol.
(1000)7 ÷ 1018 = (1000)7 / 1018 = (103)7/1018
= 10(3*7)/1018 = 1021/1018 = (10)21-18 = 103 = 1000.
 
7. (17)3.5 * (17)? = 178
 
  • A. 3.5
  • B. 4.5
  • C. 5.3
  • D. 5.4
Ans: B.
Sol.
Let (17)3.5 * (17)x = 178. Then, (17)3.5 + x = (17)8
∴ 3.5 + x = 8 ⇔ x (8-3.5) ⇔ x=4.5.
 
 
8. 49 * 49 * 49 * 49 = 7?
 
  • A. 4
  • B. 5
  • C. 7
  • D. 8
Ans: D.
Sol.
49 * 49 * 49 * 49 = (72 * 72 * 72 * 72) = 7(2+2+2+2) = 78.
So, the correct answer is 8.