Surds are numbers left in 'square root form' or 'cube root form'. A surd is the root of a whole number that has an irrational value. Some examples are ?2 ?3 ?10. All surds are irrationals but all irrational numbers are not surds. An index number is a number which is raised to a power. The power, also known as the index, tells you how many times you have to multiply the number by itself.

Surds and Indices-Surds and Indices

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Surds and Indices  Aptitude basics, practice questions, answers and explanations
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I. 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

II. Surds: Let a be a rational number and n be a positive integer such that a1/n = n√a is irrational. Then, na is called a surd of order n.

III. Laws of Surds:

1. To find √ (a + √b) write it in the form m + n + 2√mn, such that m + n = a and 4mn = b, then √ (a + √b) = ±(√m + √n)

2. (√a.√a.√a….∞) = a
3. If (√a + √a + √a……..∞) = p, then p (p – 1) = a.
4. If a + √b = c + √d, then a = c and b = d.

Examples:
1. Simplify: (i) (81)3/4        (ii) (1/64)-5/6        (iii) (256)-1/4

Solution:(i) (81)3/4 =(34)3/4 =33=27.
(ii) (1/64)-5/6 = 645/6 = (26)5/6= 25 = 32

(iii) (256)-1/4 =( 1/256)1/4 = [( 1/4)4]1/4 =1/4

2. If x=3+2√2, then the value of ( √x- (1/√x)) is:........
Solution:

Exercise Questions

1.The value of (√8)1/3 is:
a.2
b. 4
c. 2
d. 8

(√8)1/3 = (81/2)1/3= 81/6 = (23)1/6= 21/2= √2.

2. The value of 51/4 * (125)0.25 is:
a. √5
b.5√5
c.5
d.25
50.25 * (53)0.25 = 51 = 5.

3. The value of (32/243)-4/5 is:
a. 4/9
b. 9/4
c. 16/81
d. 81/16

(32/243)-4/5 = (243/32)4/5 = [(3/2)5]4/5 = 81/16

4. (1/216)-2/3 ÷ (1/27)-4/3 = ?
a. 3/4
b. 2/3
c. 4/9
d. 1/8
(1/216)-2/3 ÷ (1/27)-4/3 = 2162/3 ÷ 274/3 = (63)2/3 ÷ (33)4/3 = 4/9

5. (2n+4 -2.2n)/(2.2n+3) = 2-3 is equal to:
a. 2n+1
b. -2n+1 + 1/8
c. 9/8 - 2
d. 1

(2n+4 -2.2n)/2.2n+3  + 1/23= 7/8 + 1/8 = 1

6. If 5√5 * 53 ÷ 5-3/2 = 5a+2 , the value of a is:
a. 4
b. 5
c.6
d. 8

53/2 * 53 ÷ 5-3/2 = 5a+2

53/2 + 3 + 3/2 = 5a+2
3/2 + 3 + 3/2 = a+2
a+2=6; a=4

7. If √2n =64, then the value of n is:
a. 2
b. 4
c. 6
d. 12

√2n =64 => 2n/2 = 64= 26
n/2=6; n=12

8.The simplified form of (x7/2 /x5/2).√y3 /√y )is :
a.x2/y
b. x3/y2
c. x6/y
d. xy