Surd and Indices in Mathematics
Last Updated : 30 Aug, 2024
Indices and surds are very important ideas in math, specifically within algebra. They are useful when it comes to reducing expressions and resolving equations with different roots or powers involved. To be able to deal with various mathematical problems that range from simple algebraic ones to complex cases, it is vital to yet understand these concepts.
This paper will examine definitions, regulations, and uses of surds and indices to provide students and lovers of math with an essential foundation.
What is a Surd?
Let x be 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).
What are Indices?
- It is also known as power or exponent.
- X p, where x is a base and p is the 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 the 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 |
| 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.
Relationship between Surds and Indices
Surds and Indices are linked to powers and roots concepts in mathematics.
A surd can be represented using indices with fractional powers.
For example - ∛x can be represented as x1/3
Conclusion
Surds and Indices are mathematical concepts that are closely related in the context of powers and roots.
Surds are irrational numbers that are expressed in root form.
Indices (or exponents) represent how many times a number is multiplied by itself.
Related Articles
Practice Problems on Surd and indices in Mathematics
1. Which of the following is a surd?
a) \mathbf{\sqrt{16}}
b) \mathbf{3\sqrt{81}}
c) \mathbf{\sqrt{20}}
d) \mathbf{\sqrt{25}}
2. Simplify the following expression \mathbf{\sqrt{8}\cdot\sqrt{2}}.
3. Find the value of \mathbf{2\sqrt{18} + 3\sqrt{2}}.
4. Simplify \mathbf{\sqrt{50} / \sqrt{2}}.
5. Simplify the expression \mathbf{5^{4}\cdot5^{-2}}.
6. Simplify the expression \mathbf{(3^{2}\cdot3^{3})/3^{4}}.
7. Simplify the Indices \mathbf{(3^{2}\cdot3^{3})^{1/5}}.
8. Simplify the Indices \mathbf{27^{2/3}}.
9. Simplify the Mixed Surd \mathbf{(9/16)^{-1/2}}.
10. Simplify the Mixed Surd \mathbf{4\sqrt{2}\cdot3\sqrt{3}}.
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