Surd - SSC CGL Tier 2 - Study Material, Online Tests, Previous Year PDF

Table of Contents
1. Surds
2. Laws of Surds
3. Order and Base of a Surd
4. Pure Surds and Mixed Surds
5. Conversion of Mixed Surds into Pure Surds
6. Expressing Given Surds as Mixed Surds in Simplest Form
7. Conversion of a Surd into a Surd of Given Order
8. Conversion of Two or More Surds into Surds of the Same Order
9. Comparison of Surds
10. Addition and Subtraction of Surds
11. Multiplication of Surds
12. Division of Surds
13. Rationalising Factor
14. Conjugate Surd
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Surds

If a is a rational number and n is a positive integer such that the nth root of a, written a^1/n, is an irrational number, then a^1/n is called a surd. A surd is also called a radical of order n. The number a under the root sign is called the radicand.

Examples

i) The quantity √3 is a surd. It can be written as (3)^1/2. Here 3 is rational, 2 is a positive integer and √3 is irrational. Therefore √3 is a surd.

ii) The quantity ∛8 is not a surd. Although 8 is rational and 3 is a positive integer, ∛8 = 2 is rational; hence ∛8 is not a surd.

iii) The expression 2 + √3 is not itself a simple surd of the form a^1/n because 2 + √3 is not rational; thus it is not a surd in the strict definition above.

iv) Every surd is an irrational number, but every irrational number is not necessarily a surd of the form a^1/n with rational a and integer n.

Laws of Surds

Below are the standard algebraic rules used with surds. In each rule, assume the indicated indices are positive integers and the radicands are non-negative (as required for real principal roots).

i) For any positive rational number a and positive integer n,

ii) If n is a positive integer and a, b are non-negative rational numbers, then

iii) For positive integer n and rational a, b,

iv) If m and n are positive integers and a is a positive rational number, then

v) If m and n are positive integers and a is a positive rational number, then

Order and Base of a Surd

The order of a surd is the index of the root. For example, a surd of the form a^1/n is an nth-order surd and a is its base (radicand).

Examples:

The surd shown above is a surd of nth order with base a.

A surd with index 2 is called a quadratic surd:

A surd with index 3 is called a cubic surd:

A surd with index 4 is called a biquadratic surd:

Pure Surds and Mixed Surds

Pure surd - a surd which has no rational factor other than 1. Examples:

Mixed surd - a surd which has a rational factor other than 1 multiplied by a surd. Examples:

Conversion of Mixed Surds into Pure Surds

To convert a mixed surd into a pure surd, factor out the largest rational perfect power from the radicand so that the remaining factor under the root is not divisible by any perfect powers corresponding to the root index.

Example: Express each of the following as pure surds.

Solution:

Expressing Given Surds as Mixed Surds in Simplest Form

Example 1: Express each of the following as mixed surds in simplest form.

Solution

Conversion of a Surd into a Surd of Given Order

Method:

Obtain the order of the given surd, say n.

Obtain the order of the new surd required, say m.

Compute k = m/n (this should be an integer for a direct conversion; if not, choose an equivalent representation so that k becomes integer).

Write the new surd as (original radicand)^k/n×n/m - equivalently raise the original surd to the power k so the index becomes m.

Example 1

Convert

Solution:

n = order of given surd = 2

m = order of new surd = 6

k = m/n = 6/2 = 3

Example 2

Convert

Solution:

n = 2

m = 8

k = m/n = 8/2 = 4

Example 3: Convert

Solution:

n = 3

m = 6

k = m/n = 6/3 = 2

Example 4: Express 2 ×

Solution:

Here n = 3

m = 6

Example 5: Express

Solution:

Here n = 2

m = 4

Conversion of Two or More Surds into Surds of the Same Order

Procedure:

Let the given surds have orders n₁, n₂, n₃, ...

Find the least common multiple (LCM) of n₁, n₂, n₃, ... and denote it by n.

For each surd with order n_i, compute m_i = n / n_i.

Express each surd by raising its radicand to the power m_i and taking the nth root so that all surds become of order n.

Example 1: Convert

Solution:

n₁ = order of

n₂ = order of

n = LCM of 4 and 6 = 12

Now m₁ = 12/4

So

and

So required surds are

Example 2: Convert

Solution:

n₁ = order of

n₂ = order of

n = LCM of 2 and 3 = 6

Thus the required surds are

Comparison of Surds

If surds are of the same order, compare their radicands directly. The surd with the larger radicand is larger.

Example 3: Which surd is larger:

Solution:

Both surds are of the same order and their radicands are 26 and 35 respectively.

Since 35 > 26, the surd with radicand 35 is larger.

Example 4: Arrange the following surds in ascending order:

Solution:

The surds are of the same order, so compare radicands to arrange them in ascending order.

Comparing Surds of Distinct Orders

When surds have different orders, first convert all of them to surds of the same (smallest possible) order by taking the LCM of their orders. Then compare the resulting radicands.

Example 5: Which surd is larger:

Solution:

Orders are 3 and 4, LCM = 12.

Example 6: Which is greater:

Solution:

n₁ = 2, n₂ = 3.

n = LCM(2,3) = 6.

Example 7: Arrange the following surds in ascending order of magnitude:

Solution:

Given surds are

The orders are 2, 3, 6; LCM = 6.

Convert each to order 6 and then compare radicands.

Now compare the radicands:

Addition and Subtraction of Surds

Surds having the same irrational factor are called similar surds. Similar surds can be added or subtracted by combining their rational coefficients. For example, a√b and c√b are similar surds and can be written as (a ± c)√b.

Unlike surds (those without a common irrational factor) cannot be directly added or subtracted. To add or subtract them, express each surd in simplest form and convert them so that they share a common irrational factor.

Example 1: Simplify

Solution:

Example 2: Simplify

Solution:

Example 3: Simplify 2 ×

Solution:

Multiplication of Surds

Surds of the same order can be multiplied by multiplying their radicands and keeping the index the same. If surds are not of the same order, first convert them to surds of the same order, then multiply.

Division of Surds

Surds of the same order may be divided by dividing their radicands and keeping the order the same. If the orders differ, convert to the same order first.

Rationalising Factor

Definition: If the product of two surds is a rational number, then each of them is called the rationalising factor of the other. Rationalising the denominator means multiplying numerator and denominator by a rationalising factor so that the denominator becomes rational.

Conjugate Surd

The conjugate of a surd expression is obtained by changing the sign between two terms. Conjugate surds are used to rationalise denominators and simplify expressions. For example, if an expression is of the form (√a + √b), its conjugate is (√a - √b) and vice versa. The product of conjugate surds is usually rational.