Surds & Indices — Concept, Formulas & Examples

Free Maths notes on Surds & Indices for CAT-MBA — learn the concept, then practise.

Surds & Indices

INDICES

Indices Rules

Note: a, b, m, n ≠0, ±1

1^m = 1
0^m = 0
a¹ = a
a⁰ = 1
a^m × aⁿ = a^m+n
a^m ÷ aⁿ = a^m-n
a^m × b^m = (a × b)^m
a^m ÷ b^m = (a ÷ b)^m
$a^{1 / m}$ = $\sqrt[m]{a}$
$a^{- m}$ = $\frac{1}{a^{m}}$
${(a^{m})}^{n}$ = $a^{m \times n}$
${(a^{m})}^{\frac{1}{n}}$ = $a^{\frac{m}{n}}$
If a^m = aⁿ, then m = n
If a^m = b^m, then

a = b, if m is odd
a = ± b, if m is even

SURDS

A surd is an irrational number which is represented as the n^th root of a rational number. It is written as $\sqrt[n]{a}$ .

Therefore √5,â6,â7 are surds.

Note: â27 is not a surd as it can be simplified as 3 which is not an irrational number.

$\sqrt[n]{a}$ is read as n^th root of a, where n is the index of the surd.

â5 is read as the third root (or the cube root) of 5 and â7 is read as the fourth root of 7. If the index of the radical is not given, it is assumed to be 2.

Remember

A surd is an irrational number
A surd is of the type $\sqrt[n]{a}$ , where a is a positive rational number and n is a natural number greater than 1.

Hence, rational numbers like √9 or â125 are not surds, because these are not unresolved. √9 is equal to +3 and â125 is equal to 5.

Again, √2 × √8 is equal to √16 or +4, hence it is not a surd.

Pure Surds and Mixed Surds

In a surd of the form b $\sqrt[n]{a}$ , b is called the coefficient of the surd.

Surds with coefficient equal to 1 are called pure surds while surds with coefficients other than unity are known as mixed surds.

For example: √3,â5,â11 are pure surds whereas 2√5,3â11 and 5â7 are mixed surds.

The following operations can be performed with surds:

$\sqrt[n]{a}$ × $\sqrt[n]{b}$ = $\sqrt[n]{a \times b}$
$\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ = $\sqrt[n]{\frac{a}{b}}$
$x \times \sqrt[n]{a}$ × $y \times \sqrt[n]{a}$ = $(x + y) \times \sqrt[n]{a}$

Rationalization of Surds

Surd is an irrational number. The process of converting a surd to a rational number is called rationalization. To rationalize, we multiply the surd with a rationalizing factor. When the rationalizing factor is multiplied with the surd, we get a rational number.

For example, to rationalize √3, we multiply it with √3, (the rationalizing factor) to get √3 × √3 = 3 and to rationalize √8 we multiply it with √2 to get √8 × √2 = √16 = 4.

It is to be noted that there are always multiple rationalizing factors available. In the above example, we could have also multiplied √3 with √27 to rationalize it, since √3 × √27 = √81 = 9

Rationalizing factor of a sum of surds

Consider the term √a + √b which is the sum of two surds. Conjugate of (√a + √b) is (√a - √b).

Similarly, the conjugate of (√a - √b) is (√a + √b).

The conjugate is the rationalizing factor (RF) for a sum of surds of the form (√a + √b).

When we multiply these two terms we get, (√a + √b) × (√a - √b) = (a – b) which is a rational number.

Note: Conjugate of (a + √b) is (a - √b) and that of (a - √b) is (a + √b)

Use of Rationalization

Rationalization is mostly used to rationalize the denominator of an expression when the denominator is sum of surds.

Example

Example: Rationalize $\frac{4}{\sqrt{7} + \sqrt{3}}$

Solution:
$\frac{4}{\sqrt{7} + \sqrt{3}}$ = $\frac{4}{\sqrt{7} + \sqrt{3}}$ × $\frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} - \sqrt{3}}$

⇒ $\frac{4}{\sqrt{7} + \sqrt{3}}$ = $\frac{4 (\sqrt{7} - \sqrt{3})}{{(\sqrt{7})}^{2} - {(\sqrt{3})}^{2}}$

⇒ $\frac{4}{\sqrt{7} + \sqrt{3}}$ = $\frac{4 (\sqrt{7} - \sqrt{3})}{7 - 3}$

⇒ $\frac{4}{\sqrt{7} + \sqrt{3}}$ = √7 - √3

Example

Example: Rationalize $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$

Solution:
Given, $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$

We first multiply and divide with (√2 + √4) - √5

⇒ $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$ = $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$ × $\frac{(\sqrt{2} + \sqrt{4}) - \sqrt{5}}{(\sqrt{2} + \sqrt{4}) - \sqrt{5}}$

⇒ $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$ = $\frac{31 [(\sqrt{2} + \sqrt{4}] - \sqrt{5})}{{(\sqrt{2} + \sqrt{4})}^{2} - {(\sqrt{5})}^{2}}$

⇒ $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$ = $\frac{31 [(\sqrt{2} + \sqrt{4}] - \sqrt{5})}{1 + 4 \sqrt{2}}$

Now, the rationalizing factor will be 1 - 4√2

⇒ $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$ = $\frac{31 [(\sqrt{2} + \sqrt{4}] - \sqrt{5})}{1 + 4 \sqrt{2}}$ × $\frac{1 - 4 \sqrt{2}}{1 - 4 \sqrt{2}}$

⇒ $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$ = $\frac{31 [(\sqrt{2} + \sqrt{4}] - \sqrt{5}) \times (1 - 4 \sqrt{2})}{1^{2} - {(4 \sqrt{2})}^{2}}$

⇒ $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$ = $\frac{31 [(\sqrt{2} + \sqrt{4}] - \sqrt{5}) \times (1 - 4 \sqrt{2})}{- 31}$

⇒ $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$ = [√2 + √4 - √5](4√2 - 1)

⇒ $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$ = 8 + 4√8 - 4√10 - √2 - √4 + √5

⇒ $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$ = 8 + 8√2 - 4√10 - √2 - 2 + √5

⇒ $\frac{31}{\sqrt{2} + \sqrt{4} + \sqrt{5}}$ = 6 + 7√2 + √5 - 4√10

Comparison of Surds

It is difficult to compare two surds of different indices. The strategy we follow is to change both surds to the same order. We can then compare them by the value of their radicands.

The new index of the two surds is the LCM of the original indices of the surds. The following example should make it clear.

Example

Example: Compare √2 and â3

Solution:
The order of the surds is 2 and 3, hence we convert both the surds to surds of order 6 (which is the LCM of 2 and 3).

√2 = $2^{1 / 2}$ = ${(2^{3})}^{1 / 6}$ = $8^{1 / 6}$

and, â3 = $3^{1 / 3}$ = ${(3^{2})}^{1 / 6}$ = $9^{1 / 6}$

Now since, 8 < 9, we can say that $8^{1 / 6}$ < $9^{1 / 6}$

Hence, √2 < â3

Calculating Square Root of Surds

If there exists a square root of a surd of the type a + √b , then it will be of the form √x + √y . We can equate the square of √x + √y to a + √b and then solve for x and y.

Note: When there is an equation with rational and irrational terms, the rational part on the left hand side is equal to the rational part on the right hand side and, the irrational part on the left hand side is equal to the irrational part on the right hand side of the equation.

Example

Example: Calculate square root of 5 + 2√6

Solution:
Square root of 5 + 2√6 will be of the form √x + √y.

∴ (√x + √y)² = 5 + 2√6

⇒ x + y + 2√xy = 5 + 2√6

Now, rational part of RHS = rational part on LHS
⇒ x + y = 5 …(1)

Also, irrational part of RHS = irrational part on LHS
⇒ 2√xy = 2√6
⇒ xy = 6 …(2)

Solving (1) and (2), we get
x = 3 and y = 2 or x = 2 and y = 3.

In both cases square root of 5 + 2√6 = √2 + √3

Example

Example: Calculate positive square root of 5 - 2√6

Solution:
Square root of 5 - 2√6 will be of the form √x - √y.

∴ (√x - √y)² = 5 - 2√6

⇒ x + y - 2√xy = 5 - 2√6

Now, rational part of RHS = rational part on LHS
⇒ x + y = 5 …(1)

Also, irrational part of RHS = irrational part on LHS
⇒ - 2√xy = - 2√6
⇒ xy = 6 …(2)

Solving (1) and (2), we get
x = 3 and y = 2 or x = 2 and y = 3.

The square root of 5 - 2√6 will be either √3 - √2 or √2 - √3

We will reject √2 - √3 as it is negative

∴ Square root of 5 - 2√6 is √3 - √2

If there exists a square root of a surd of the type a + √b + √c + √d , then it will be of the form √x + √y + √z. [x, y, z ≥ 0]

Example

Example: Calculate square root of 6 + 2√2 + 2√3 + 2√6

Solution:
Square root of 6 + 2√2 + 2√3 + 2√6 will be of the form √x + √y + √z.

∴ (√x + √y + √z)² = 6 + 2√2 + 2√3 + 2√6

⇒ (x + y + z) + 2√xy + 2√xz + 2√yz = 6 + 2√2 + 2√3 + 2√6

⇒ x + y + z = 6 …(1)
⇒ xy = 2 …(2)
⇒ xz = 3 …(3)
⇒ yz = 6 …(4)

⇒ x = 1, y = 2 and z = 3

∴ Square root of 6 + 2√2 + 2√3 + 2√6 will be √1 + √2 + √3 = 1 + √2 + √3

Video Concepts - Surds & Indices

Hello MBA Aspirants,

Indices & Surds is an important topic for CAT and other management entrance exams. Here, you'll find Concept Videos for Indices & Surds. Apart from videos you'll find Concept Review Exercise (CRE) after each video. Once you go through each video, you should practice respectively CRE to understand the concept well.

Once, you're done with all videos and CREs, you should attempt advance level questions given in the Practice Exercises. For additional practice you can also practice Previous Year Questions from CAT and OMETs given at the end.

All the Best for your preparation !!!

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Surds & Indices — Concept, Formulas & Examples

Surds & Indices

INDICES

Indices Rules

SURDS

Pure Surds and Mixed Surds

Rationalization of Surds

Comparison of Surds

Calculating Square Root of Surds

Video Concepts - Surds & Indices

INDICES

SURDS

SURDS BASICS & OPERATIONS

CALCULATING SQUARE ROOT OF A SURD

COMPARISON OF A SURD

PRACTICE EXERCISES

PREVIOUS YEAR QUESTIONS

Practise Surds & Indices

Track your progress on Surds & Indices