Quantitative Aptitude-Surds and Indices updated on Nov 2021
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.

Quantitative Aptitude

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

Answer: Option c.
(√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
Answer: Option c
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

Answer: Option d.
(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
Answer: Option c.
(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
Answer: Option d.

(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

Answer: Option a
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

Answer: Option d
√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
Answer: Option d

(x7/2 /x5/2 ). (√y3 /√y) = x7/2 -5/2. y3/2 - 1/2 = xy

 

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. Freshersworld.com explains basic concept of Surds and indices and provides aptitude test, shortcuts, tricks, and formulas. It also provides questions and answers, practice questions and solved examples that would help candidates in clearing all the competitive tests. It also provides online test on Surds and indices - quantitative Aptitude. What are the rules of a surd? Some of the important rules are mentioned below • Every rational number is not a surd. • Every irrational number is a surd. • A root of a positive real quantity is called a surd if its value cannot be exactly determined. • ?a × ?a = a ? ?5 × ?5 = 5 • If a and b are both rationals and ?x and ?y are both surds and a + ?x = b + ?y then a = b and x = y • If a – ?x = b – ?y then a = b and x = y. • If a + ?x = 0, then a = 0 and x = 0. • If a – ?x = 0, then a = 0 and x = 0. What are the rules of indices? a0=1 a-m=1/am am*an=am+n am/an=am-n (am)n=amn a1/a1=a0
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