Surd and indices in Mathematics
Surds :
Let x is a rational number(i.e. can be expressed in p/q form where q ≠0) and n is any positive integer such that x1/n = n √x is irrational(i.e. can’t be expressed in p/q form where q ≠0), then that n √x is known as surd of nth order.
Example –
√2, √29, etc.
√2 = 1.414213562…, which is non-terminating and non-repeating, therefore √2 is an irrational number. And √2= 21/2, where n=2, therefore √2 is a surd. In simple words, surd is a number whose power is an infraction and can not be solved completely(i.e. we can not get a rational number).
Indices :
- It is also known as power or exponent.
- X p, where x is a base and p is power(or index)of x. where p, x can be any decimal number.
Example –
Let a number 23= 2×2×2= 8, then 2 is the base and 3 is indices.
- An exponent of a number represents how many times a number is multiplied by itself.
- They are used to representing roots, fractions.
Rules of surds :
When a surd is multiplied by a rational number then it is known as a mixed surd.
Example –
2√2, where 2 is a rational number and √2 is a surd. Here x, y used in the rules are decimal numbers as follows.
S.No. |
Rules for surds |
Example |
1. |
n √x = x1/n |
√2 = 21/2 |
2. |
n√(x ×y) =n √x × n √x |
√(2×3)= √2 × √3 |
3. |
n√(x ÷y)=n √x ÷ n √y |
3√(5÷3) = 3√5 ÷ 3√3 |
4. |
(n √x)n = x |
(√2)2 = 2 |
5. |
(n√ x)m = n√(x m) |
(3√27)2 = 3√(272) = 9 |
6. |
m√(n√ x) = m × n √x |
2√(3√729)= 2×3√729 = 6√729 = 3 |
Rules of indices :
S.No. |
Rules for indices |
Example |
1. |
x0 = 1 |
20 = 1 |
2 |
x m × x n = x m +n |
22 ×23= 25 = 32 |
3 |
x m ÷ x n = x m-n |
23 ÷ 22 = 23-2 = 2 |
4 |
(x m)n = x m ×n |
(23)2 = 23×2 = 64 |
5 |
(x × y)n = x n × y n |
(2 × 3)2= 22 × 32 =36 |
6 |
(x ÷ y)n = x n ÷ y n |
(4 ÷ 2) 2= 42 ÷ 22 = 4 |
Other Rules :
Some other rules are used in solving surds and indices problems as follows.
// From 1 to 6 rules covered in table.
7) x m = x n then m=n and a≠0,1,-1.
8) x m = y m then
x = y if m is even
x= y, if m is odd
Basic problems based on surds and indices :
Question-1 :
Which of the following is a surd?
a) 2√36 b) 5√32 c) 6√729 d) 3√25
Solution –
An answer is an option (d)
Explanation -
3√25= (25)1/3 = 2.92401773821... which is irrational So it is surd.
Question-2 :
Find √√√3
a) 31/3 b) 31/4 c) 31/6 d) 31/8
Solution –
An answer is an option (d)
Explanation -
((3 1/2)1/2) 1/2) = 31/2 × 1/2 ×1/2 = 3 1/8 according to rule number 5 in indices.
Question-3 :
If (4/5)3 (4/5)-6= (4/5)2x-1, the value of x is
a) -2 b)2 c) -1 d)1
Solution –
The answer is option (c)
Explanation -
LHS = (4/5)3 (4/5)-6= (4/5)3-6 = (4/5)-3
RHS = (4/5)2x-1
According to question LHS = RHS
⇒ (4/5)-3 = (4/5)2x-1
⇒ 2x-1 = -3
⇒ 2x = -2
⇒ x = -1
Question-4 :
34x+1 = 1/27, then x is
Solution –
34x+1 = (1/3)3
⇒34x+1 = 3-3
⇒4x+1 = -3
⇒4x= -4
⇒x = -1
Question-5 :
Find the smallest among 2 1/12, 3 1/72, 41/24,61/36.
Solution –
The answer is 31/72
Explanation –
As the exponents of all numbers are infractions, therefore multiply each exponent by LCM of all the exponents. The LCM of all numbers is 72.
2(1/12 × 72) = 26 = 64
3(1/72 ×72) = 3
4(1/24 ×72) = 43 = 64
6 (1/36 ×72) = 62 = 36
Question-6 :
The greatest among 2400, 3300,5200,6200.
a) 2400 b)3300 c)5200 d)6200
Solution –
An answer is an option (d)
Explanation –
As the power of each number is large, and it is very difficult to compare them, therefore we will divide each exponent by a common factor(i.e. take HCF of each exponent).
The HCF of all exponents is 100.
2400/100 = 24 = 8.
3300/100 = 33 = 27
5200/100 = 52 = 25
6200/100= 62 = 36
So 6200 is largest among all.
Last Updated :
15 Jun, 2021
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