Surd and Indices in Mathematics

Surds are irrational numbers that are expressed in root form, such as √2 or √5, and cannot be simplified into whole numbers.

Indices are used to represent repeated multiplication of a number by itself. They show the power or exponent of a number, such as 2³, where 2 is the base, and 3 is the index.

Note: 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 x^1/3

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 x^1/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 a 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= 2^1/2, where n=2, therefore √2 is a surd. In simple words, a surd is a number whose power is an infraction and can not be solved completely(i.e., we can not get a rational number).

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 and y used in the rules are decimal numbers as follows.

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 Indices

When a number is expressed in exponential (power) form, it is said to be written using indices.

Example
2³, where 2 is the base and 3 is the index.

Here, x and y used in the rules represent numbers, and m and n represent integers.

Other Rules

Some other rules are used in solving surds and indices problems as follows.

From 1 to 6 rules covered in table.
7) If xᵐ = xⁿ, then m = n (when x ≠ 0, 1, -1)
8) x ^m = y ^m then
x = y, if m is odd
x = ±y if m is even

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) 3^1/3
b) 3^1/4
c) 3^1/6
d) 3^1/8

Solution:

An answer is an option (d) 

Explanation: ((3 ^1/2)^1/2) ^1/2) = 3^{1/2 × 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: 3^4x+1 = 1/27, then x is

Solution:

3^4x+1 = (1/3)³ ⇒3^4x+1 = 3^-3 ⇒4x+1 = -3 ⇒4x= -4 ⇒x = -1

Question 5: Find the smallest among 2 ^1/12 , 3 ^1/72, 4^1/24, 6^1/36.

Solution:

The answer is 3^1/72

Explanation:
As the exponents of all numbers are in fractions, 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 2^400, 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. 2^400/100 = 2⁴ = 8. 3^300/100 = 3³ = 27 5^200/100 = 5² = 25 6^200/100= 6² = 36 So 6²⁰⁰ is largest among all.

Practice Problems on Surd and indices

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}}.

Related Articles

• Power in Mathematics
• Exponents
• Adding and Subtracting Exponents
• How to Multiply and Divide Exponents?