Surds and Indices

Indices :-

When a number 'a' is multiplied by itself 'n' times, then the product is called nth power of 'p' and is written as aⁿ. Here, a is called base and 'n' is known as the exponent of the power.
Therefore, aⁿ is the exponential expression. aⁿ is read as 'p raised to the power n'.

Rules of Indices :-

1.a^m × aⁿ = a^m+n

Ex- 2³ × 2² = ?
Solution:- 2³ × 2²
= 2^{3 + 2} = 2⁵ = 32

2. a^m ÷ aⁿ = a^{m - n}

Ex- 2⁹ ÷ 2³ = ?
Solution:- 2⁹ ÷ 2³
= 2^{9 - 3}
= 2⁶ = 64

3. ( a^m )ⁿ = a^m×n

Ex- ( 3¹⁵)^1/5 = ?
Solution:- = ( 3¹⁵)^1/5
= 3^{15× ( 1/5 )}
= 3³
= 27

4. ( a )⁰ = 1

Ex- 342⁰ = 1
432⁰ = 1
7445⁰ = 1

5. ( ab )ⁿ = aⁿ × bⁿ

Ex- ( 5 × 3 )³
= 5³ × 3³
= 125 × 27
= 3375

6. a ^m = a^m

b b^m

Ex-		5		⁴	= ?
		3

Solution:- Using above given rule ,		5		⁴	=	5⁴
		3				3⁴

=	625
	81

7. ( a )^-m = 1

a^m

Ex- ( 7 )^-3 =	1
	7³

=	1
	343

Ex- Simplify ( 256 )^3/4
Sol:- ( 256 )^3/4
= ( 4⁴ )^3/4
= 4^4×3/4
= 4³ = 64

Ex- Simplify ( 512 )^-3/9
Sol:- ( 512 )^-3/9
= ( 2⁹ )^-3/9
= 2^9×-3/9

= 2^-3 =	1
	2³

=	1
	8

Surds :-

Surds are numbers left in root form does not provide an exact solution,then this is called a surd.
It has an infinite number of non-recurring decimals.
Therefore, surds are irrational numbers.
For example √2, √3 , 5√2 etc.

Order of Surds :-

Let a be a rational number and m be a positive integer such that a^1/m = ^m√a is irrational.
Then, ^m√a is called a surd on mth order and a is called the radicand.
Simplify, 5^1/2 = √5 = surd of 2nd order.
11^1/5 = ⁵√11 = surd of 5th order.
surds of order 2 are known as quadratic surds.
surds of order 3 are known as cubic surds.

Rules of Surds :-

1. ^m√a = a^1/m

Ex- ⁵√7 = 7^1/5 , ³√13 = 13^1/3

2. ^m√ab = ^m√a × ^m√b

Ex- ⁷√3×13 = ⁷√3 × ⁷√13

3. ^m√ a = ^m√a

b ^m√b

Ex- ⁵√	7	=	⁵√7
	13		⁵√13

4. ( ^m√a ) ^m = a

Ex- ( ⁷√242 ) ⁷ = 242

5. ( ^m√a )ⁿ = ( a^1/m )ⁿ = a^n/m = ^m√aⁿ

Types of Surds :-

Pure Surds :-

A Surd which do not have factor other than 1, is called pure surds. For example √2, ⁷√5 etc.

Mixed Surds :-

A surds which have factor other than 1, is called mixed surds.
For example 4 √2, 6√7 etc.

Like Surds :-

When the radicands of two surds are same, then those are known as like surds. For example 6√7, 3√7 etc.

Unlike Surds :-

When radicands of two surds are different, then they are known as unlike surds.
For example 6√7, √3 etc.

Properties of Surds :-

A quadratic surd can not be equal to the sum and difference of a rational number and a quadratic surd.
For example a + b ≠ √c or √a - b ≠ √c
If a + √b = c + √d or a - √b = c - √d, than a = c and b = d.
If a + √b = c + √d, then a - √b = c - √d and vice-versa.
If √a +√b = √c + √d, then √a - √b = √c - √d and its vice-versa.

Important Point :-

x^b

x^c

x^a

= 1

x^c

x^a

x^b

Proof:-

x^b

x^c

x^a

x^c

x^a

x^b

= ( x^b-c )^a ( x^c-a )^b ( x^a-b )^c
= x^ab-ac x^bc-ba x^ac-bc
= x^{ab - ac + bc - ba + ac - bc}
= x⁰ = 1

x^a

^{a + b}

x^{b + c}

x^c

^{c + a}

= 1

x^b

x^c

x^a

( ^{a² + b² + ab} )

x^b

( ^{b² + c² + bc} )

x^c

( ^{c² + a² + ca} )

= 1

x^b

x^c

x^a

To Arrange the Surds in Increasing or Decreasing order

Let given surds are x^1/p, y^1/q, z^1/r
To arrange the surds in increasing or decreasing order first take the LCM of p, q and r and use it to make the denominator of the powers the same. Then easily can find the required order.

Ex- Arrange √2, ³√5 and ⁴√3 in ascending order.
Solution:- √2, ³√5 and ⁴√3
= 2^1/2, 5^1/3 and 3^1/4
So, p = 2 , q = 3 , and r = 4
LCM of 2, 3 and 4 = 12
Then, we will make the denominator of the power of every coefficient equal i.e, 12.
√2 = 2^1/2 = 2^6/12 = ¹²√64
³√5 = 5^1/3 = 5^4/12 = ¹²√625
⁴√3 = 3^1/4 = 3^3/12 = ¹²√27
²√27 < ¹²√64 < ¹²√625
⁴√3 < √2 < ³√5

Operation on Surds

Addition of surds

To add the surds first simplify them and add the like surds only like surds can be added.
√a + √b ≠ √ ( a + b )
x √a + x √b ≠ x √ ( a + b )

Ex- Find the value 3 √150 + 2 √216 + 4 √54
Solution:- 3 √150 = 3 √ ( 25 × 6 ) = 15 √6
2 √216 = 2 √ ( 36 × 6 ) = 12 √6
4 √54 = 4 √ ( 9 × 6 ) = 12 √6
= 15 √6 + 12 √6 + 12 √6
= 39 √6

Subtraction of surds

Only like surds can be subtracted. Therefore, to subtract two or more surds,first simplify them and subtract them.
√a - √b ≠ √ ( a - b )

Ex- Find the value of 7 √405 - 2 √20 - √45
Solution:-
7 √405 = 7 √ ( 81 × 5 ) = 63 √5
2 √20 = 2 √ ( 4 × 5 ) = 4 √5
√45 =√ ( 9 × 5 ) = 3 √5
= 63 √5 - 4 √5 - 3 √5
= 56 √5

Multiplication and Division of surds

To multiply or divide the surds, we make the denominators of the powers equal to each other. Then multiply or divide as usual.

Ex- Find the product of √2, ³√3 and ²√4
Solution:- LCM of 3 and 2 = 6
√2 = 2^1/2 = 2^3/6 = 8^1/6
³√3 = 3^1/3 = 3^2/6 = 9^1/6
²√4 = 4^1/2 = 4^3/6 = 64^1/6
= ( 8 × 9 × 64 )^1/6
= ( 4608 )^1/6
= ⁶√4608

Comparision of Surds

To compare two or more surds, first of all the denominators of the power of given surds are made eqal to each other and then the redicand of the new surds are compared.

Ex- Which of the following surds is greatest?
√2, ⁶√3, ³√4 , and ³√5
Solution:-
√2 = 2^1/2 , ⁶√3 = 3^1/6 , ³√4 = 4^1/3 , ³√5 = 5^1/3
LCM of 2, 3 and 6 = 6
2^1/2 = 2^3/6 = 8^1/6
3^1/6 = 3^1/6 = 3^1/6
4^1/3 = 4^2/6 = 16^1/6
5^1/3 = 5^2/6 = 25^1/6
∴ ³√5 is the greatest of all the given surds.

Rationalisation of Surds

The method of obtaining a rational number from a surd by multiplying it with another surd is known as rationalisation of surds. Both the surds are known as rationalising factor of each other.

Ex- Find the fraction equivalent to	7
	4 - √15

such that denominator of the fraction is not irrational.

Solution:-	7
	4 - √15

=	7	×	4 + √15
	4 - √15		4 + √15

=	{ 7 × ( 4 + √15 ) }
	[ 4² - ( √15 )² ]

=	{ 7 × ( 4 + √15 ) }
	( 16 - 15 )

= 7 ( 4 + √15 )

Ex- If x =	√3 + 1	and y =	√3 - 1
	√3 - 1		√3 + 1

then ,find the value of x² + y².

Solution- x =	√3 + 1	×	√3 + 1
	√3 - 1		√3 + 1

=	( √3 + 1 )²
	( √3 )² - ( 1 )²

=	( 3 + 1 + 2√3 )
	( 3 - 1 )

=	( 4 + 2√3 )
	2

=	2( 2 + √3 )
	2

x = 2 + √3

y =	√3 - 1	×	√3 - 1
	√3 + 1		√3 - 1

=	( √3 - 1 )²
	( √3 )² - ( 1 )²

=	( 3 + 1 - 2√3 )
	( 3 - 1 )

=	( 4 - 2√3 )
	2

=	2( 2 - √3 )
	2

y = 2 - √3
x + y
= 2 + √3 + 2 - √3 = 4
xy = ( 2 + √3) × ( 2 - √3 )
= 2² - (√3)²
= 4 - 3
= 1
Now, x + y = 4
Squaring on both sides
or, ( x+y )² = 4²
or, x² + y² + 2xy = 16
or, x² + y² + 2 × 1 = 16
or, x² + y² + 2 = 16
or, x² + y² = 16 - 2
∴ x² + y² = 14

Some Important Rule :-

Rule- √ ( x√x√x√x…….∞ ) = x

Ex- √ ( 7√7√7√7…….∞ )
Solution:- Let x = √ ( 7√7√7√7…….∞ ) x = √7x
Squaring on both sieds
or, x² = 7x
∴ x = 7

Rule- √ ( x + √x + √x + √x + …….∞ )
To solve this type of question, first find the factors of x, if the factors of x is two consecutive numbers then the greatest factor is the answer.
If the factors of x is not two consecutive numbers then answer will be

√( 4x + 1 ) + 1

2

Ex- Find the value of √ ( 72 + √72 + √72 + √72 + …….∞ )
Solution:- 72 = 9 × 8
∴ √ ( 72 + √72 + √72 + √72 + …….∞ ) = 9

Ex- Find the value of √ ( 77 + √77 + √77 + √77 + …….∞)
Solution:- Here, x = 77 = 7 × 11
7 and 11 are not consecutive

∴√ ( 77 + √77 + √77 + √77 + …….∞ ) =	√( 4x + 1 ) + 1
	2

	√( 4 × 77 + 1 ) + 1
	2

	√( 309 + 1 )
	2

Rule- √ ( x - √x - √x - √x - …….∞ )
To solve this type of question, first find the factors of x, if the factors of x is two consecutive numbers then the lowest factor is the answer.
If the factors of x is not two consecutive numbers then answer will be

√( 4x + 1 ) - 1

2

Ex- Find the value of √ (42 - √42 - √42 - √42 - …….∞ )
Solution:- Here, x = 42 = 6 × 7
∴ √ ( 42 - √42 - √42 - √42 - …….∞ ) = 6

Ex- Find the value of √( 35 - √35 - √35 - √35 - …….∞ )
Solution:- Here, x = 35 = 5 × 7

∴√35 - √35 - √35 - √35 - …….∞ =	√( 4x + 1 ) - 1
	2

=	√( 4 × 35 + 1 ) - 1
	2

=	√( 141 - 1 )
	2

Rule- √ (x√x√x√x ) = x^{( 2ⁿ - 1)/2ⁿ}

Ex- Find the value of √ (13√13√13√13 )
Solution:- Here, n = 4
∴ √ (13√13√13√13 ) = 13^{( 2ⁿ - 1)/2ⁿ}
= 13^{( 2⁴ - 1)/2⁴}
= 13^15/16