Pascal's Triangle & Binomial Theorem

Radicals and Exponents

Masuma Parvin Senior Lecturer Department of GED

Daffodil International University

2

Definition

An expression containing the radical symbol is called a radical.

 

The general form of a radical is , where is the index and is the radicand.

 

If is any real number and is a positive integer, then the product of numbers is defined as

 

� ^�

 

Base

(real number)

Exponent (integer)

� √ ^�

 

Index Radical Sign

Radicand

Exponent

Radical

Where is the index or exponent or power and is the base.

 

Note:

If omit the index and write rather than .

 

Properties of Exponents

Let and be real numbers, variables, or algebraic expressions and let and be integers (All denominators and bases are nonzero).

 

Property Example

( _�

²

₊ 1 )

⁰

=1, ( _�

²

+1 )

¹

_{= �}

²

₊ 1

�

⁰

= 1, �

¹

= �

 

1.  

3²

3⁴ =3²⁻⁴=3^{− 2}= 1

3² = 1 9

�^�  

�^� = �^{� − �} 4.  

(

�³

)

⁻⁴=�³⁽⁻⁴⁾=�⁻¹²= 1

�¹²

( _�

^�

)

^�

_{= �}

^��  

5.  

3

²

3

⁴

= 3

²⁺⁴

= 3

⁶

=729   �

^�

�

^�

= �

^�+�  

3.  

(

�

² )

³⁼

²

_�^{33 =} _�

⁸

³

(

^��

)

^{� = �}_�^��  

7.  

3^{− 4} = 1

3⁴ =

(

¹3

)

⁴

�^−�= 1  

�^� =

(

�¹

)

^�

2.  

( 5 � )

³

= 5

³

�

³

= 125 �

³

( �� )

^�

=�

^�

�

^�  

6.  

3

Properties of Radicals

Let and be real numbers, variables, or algebraic expressions such that the indicated roots are real numbers, and let and be positive integers.

 

Property Example

√

3

⁸

²

^{= (}

³

^{√ 8 )}

²

^{= ( 2 )}

²

⁼⁴

�  

√ ^�

^�

^{= (}

^�

^{√ � )}

^�

1.  

√ ^{35 =} √ ^{5 ⋅ 7 =} √ ^{5 ⋅} √ ⁷

�

 

√ ^{�� = �} √ ^{�⋅ �} √ ^�

4.  

√

3

^{27 8 = √}

³

^√

³

^{27 8 = √}

³³

^{√ 3 2}

³³

^{= 3 2}

�  

√ ^{� � =}

^��

^{√ √ � � , � ≠ 0}

 

5.

√

3

^{( − 12)}

³

^{= − 12}

For �   odd ,  

^�

√ ^�

^�

^{= �}

 

3.  

√

3

^√

⁵

¹⁰⁼

^(3×5

^√

⁾

¹⁰⁼

^{1 5}

^{√ 10}

m  

√

^�

^{√ � =}

^{m n}

^{√ a}

6.  

√

4

^{( − 12)}

⁴

^{= | − 12 | = 12}

Foreven,  

2.  

4

Simplification of Radicals

An expression involving radicals can be simplified by,

1. by removing the perfect nth powers of the radicand.

2. by reducing the index of the radical

3. by rationalizing of the denominator of the radicand.

Find the simplest form of the followings:

a. b. c. d. e.

 

a. We have

  Property 3

b. We have

  Property 4

c. We have

  Property 5

d. We have

  Property 1

e. We have

  Property 6

5

Find the simplest form of the followings:

 

We have, a.

b.

c. d.

¿

 

√

⁶

^{( 9 � )}

²

¿

(

( 9�)²

)

1

  6

¿(9�)

1

3 =

√

^{3 9�}

 

We have,

 

¿ √

⁶

⁹

²

^�

²

 

¿  49⋅

( √

^{3 4 ��}

⁾

²

¿ 

98

⋅³

√ ²

^�2�2

√

 3

^{64 �}

⁷

^�

⁻⁶

¿ 

√

^{3 43 ⋅}

√

³

⁽

^�2

⁾

^{3 ⋅}

^√

^{3 � ⋅}

^√

^{3 �−6}

¿  4⋅ �²⋅

√

^{3 � ⋅ �−2}

¿ 4 �²

�²

√

3 ^�

 

We have,

√

3

^{( (}

^��−+

^{1 2 ) )}

³⁶

 

¿

√

3 ^{(� +1)3}

√

3 ^{( � −2)6}

 

¿ � +1 ( � −2)²

 

We have,

(

 7 ³

√

^4��

⁾

²

¿ 

49

⋅

√

³

²

³

^.2

^�2�2

6

¿  49⋅

√

^{3 (4��)2}

¿

 

√

³

⁴

³

^{⋅ �}

⁶

^{⋅ � ⋅ �}

⁻⁶

7

Calculate the followings:

 

√

 

^{18 +} √ ^{50 −} √ ⁷²

¿ 

√ ²

^⋅

³

²⁺

√ ²

^⋅

⁵

²⁻

√ ²

^3⋅

³

²

¿

 

3 √ ^{2 +5} √ ^{2 − 3} √ ²

²

^{⋅ 2}

¿

 

3 √ ^{2 + 5} √ ^{2 − 6} √ ²

¿

 

2 √ ²

2

 

√ ^{27 − 4} √ ¹²

¿ 

2 √ ³

^2⋅

³

⁻

⁴ √ ²

^2⋅

³

¿

 

2 ⋅ 3 √ ^{3 − 4 ⋅ 2} √ ³

¿

 

6 √ ^{3 − 8} √ ³

¿

 

− 2 √ ³

a.

b.

√

 

^{248 + √ 52+ √ 144}

¿

 

√ ^{248 + √ 52 + √ 2 4 ⋅ 3 2}

¿

 

√ ^{248 + √ 52 + 2}

²

^{⋅ 3}

¿

 

√ ^{248 + √ 52 + 12}

¿

 

√ ^{248 + √ 64}

¿

 

√ ^{248 +} √ ²

⁶

¿ 

√ ²⁴⁸

⁺

²

³

¿

 

√ ^{248+ 8}

¿ 

√ ²

^8=24=16

d.

112

√

^{196 ×}

√

⁵⁷⁶

12 ×

√

²⁵⁶

8

 

¿ 112

√

^{22⋅72 ×}

√

^26⋅32

12 ×

√

²⁸

8

 

¿ 112

2⋅7 × 2³⋅3

12 × 2⁴ 8

 

¿ 112

2⋅7 × 8⋅3

12 × 16

8 =32

 

c.

¿

 

32

Show that

 

¿ 3 + √ ³

3 + ( 3 − √ ^{3 )}

3

²

− ( √ ^{3 )}

²

⁻

( 3 + √ ^{3 )}

3

²

− ( √ ^{3 )}

²

 

¿ 3 + √ ³

3 + ( _{3 −} √ ^{3 )}

9 − 3 − ( _{3 +} √ ^{3 )}

9 − 3

 

L . H . S .= 18 + 2 √ ^{3 +3 −} √ ^{3 − 3 −} √ ³

6

 

¿ 18

6 =3

 

¿ 3 + √ ³

( √ ^{3 ) (} √ ^{3 ) +}

( _{3 −} √ ^{3 )}

( 3 + √ ^{3 ) ( 3 −} √ ^{3 ) −}

( _{3 +} √ ^{3 )}

( 3 − √ ^{3 ) ( 3 +} √ ^{3 )}

 

L . H. S   =3 + 1

√ ^{3 +}

1

3 + √ ^{3 −}

1 3 − √ ³

 

8

= 3 + √ ³

3 + ( _{3 −} √ ^{3 )}

6 − ( _{3 +} √ ^{3 )}

6

 

9

�+�

√

^{6= 7}

√

³⁺⁵

√

²

√

⁴⁸⁻

√

¹⁸

 

¿ 7

√

³⁺⁵

√

²

4

√

³⁻³

√

²

 

¿

(

7

√

³⁺⁵

√

²

^{) (}

⁴

√

³⁺³

√

²

⁾

(

4

√

³⁻³

√

²

^{) (}

⁴

√

³⁺³

√

²

⁾

 

¿ 28×3+21

√

²

√

³⁺²⁰

√

²

√

^3+15×2

48−18

 

¿ 84+21

√

⁶⁺²⁰

√

⁶⁺³⁰

30

 

¿ 114+41

√

⁶

30

 

�� ,�+�

√

⁶⁼¹¹⁴

30 + 41 30

√

⁶

 

∴�= 114

30 =19

5       and  �= 41 30

 

Find the value of & if

 

Given that,

What will be come in the place of question mark

 

√ ^{86.49 +} √ ^{5 + ( � ) =12.3}

 

o

 

r ,     √ ^{5 + ( � ) =12.3 −} √ ^86.49

 or ,  5 + � = ( ¹²³ 10 − √ ^{8649 100 )}

²

 

or , 

 

� = ( ^{123 10 − √} _√ ^{8649 100} )

²

^{− 5}

or ,  � = ( ^{123 10 − 93 10} )

²

^{− 5}

 

or ,     � = ( ^{30 10} )

²

^{− 5}

 

or ,  � = ( 3 )

²

− 5

 

  or ,  � =9 − 5

 

  or ,  � =4

 

Let the required value is

 

According to the question we can write,

10

What will be come in the place of question mark

 

( � )

1

4= 48 ( � )

3 4

 

o r , (�)

1

4 ( �)

3

4 =48

 

o

r , (

�

)

1 4 + 3

4 =

48

 

o

r , (

�

)

4

4 =

48

 

Let the required value is

 

According to the question we can write,

∴ � = 48

 

11

If then find the value of

 

Given that,

 

√ ^{841 +} √ ^{8.41 +} √ ^{0.0841 +} √ ^0.000841

 

¿ √ ^{841 +} √ ^{841 100 +} √ ^{10000 841 +} √ ^{1000000 841}

 

¿ √ ^{841 +} √ ⁸⁴¹

√ ^{100 +}

√ ⁸⁴¹

√ ^{10000 +}

√ ⁸⁴¹

√ ^1000000

 

¿ 29 + 29

10 + 29

100 + 29 1000

 

¿ 29000 + 2900 + 290 + 29 1000

 

¿ 32219 1000

 

¿

 

32.219

Now,

12

If, then find the value of

 

We have,

√ ¹

⁺

¹⁴⁴

^{� =}

^{13 12}

 

or ,  1 + �

144 = ( ^{13 12} )

²

 

  or ,  �

144 = 169

144 − 1

 

  or ,  �

144 = 169 − 144 144

 

or ,  �

144 = 25 144

 

  ∴ � =25

 

13

14

(243)

�

5 ×3^2�+1 9^�×3^�−1

 

¿ (3)^{5⋅ �5} × 3^2�+1 3^2�× 3^�−1

 

¿ 3^�×3^2�+1 3^2�+�−1

 

¿ 3^�+2�+1 3^2�+�−1

  ¿ 3^3�+1

3^3�−1

 

¿ 3

^{3�+1−3�+1}

 

¿ 9

 

Find the value of

 

We have,

¿ 3

²

 

15

If , then find the value of

 

Given that,  

2

^�

+ 2

^{1− �}

= 3

  or ,  2

^�

+ 2

¹

⋅ 2

^−�

= 3

 

  or ,  2^�+2⋅ 1

2^� =3

 

  or ,  � + 2 ⋅ 1

� =3

 

  or ,  �

²

+ 2= 3 �

 

  or ,  �

²

− 3 � + 2= 0

 

  or ,  �

²

− 2 � − �+ 2 =0

 

or ,  � ( � − 2 ) − 1 ( � − 2 ) =0

 

or ,  ( � − 1 )( � − 2 ) =0

 

Therefore   � − 1 =0 ∧ � − 2= 0

 

⇒

 

  �=1

⇒   2

^�

= 2

⁰

 

⇒   � = 0

 

⇒� =2

 

⇒ 2

^�

=2

¹

 

⇒ � =1

 

Let    2

  ^�

=�

16

If and , then what is the value of and ?

 

Given that,  

8

^�

. 2

^�

= 512

  or ,  ( 2

³

)

^�

. 2

^�

=2

⁹

 

or ,  2

^3�

. 2

^�

=2

⁹

   

� = 3

or ,  2

^3�+�

= 2

⁹

 

∴

 

  3 � + � =9 …… … ( � )

Again,  

3

^{3 �+2 �}

=9

⁶

or , 3

^{3 �+2 �}

=( 3

²

)

⁶

 

or , 3

^{3 �+2 �}

= 3

¹²

 

∴

 

 3 � + 2 � =12 ……… ( �� )

Subtracting equation from equation we get,

 

Putting the value of in equation we get,

 

  3 � + 3 = 9

 

or , 3 � =9 − 3 = 6

 

∴   � =2

 

T herefore ,   � = 2∧ � = 3

 

Exercise

17

1. Find the simplest form of the followings:

2. Evaluate by factorization method.

3. Show that .

4. Find the cube root of .

5. If and then find the value of . 6. If then find the values of . 7. If then what are the values of ? 8. Find the values of