# CLASS-7PROBLEM & SOLUTION OF ALGEBRAIC EXPONENTS

PROBLEM & SOLUTIONS EXAMPLES

Express each of the following as a power of a number with a positive index –

1)  2⁴ X 64 X 2² = ?

Ans.)   2⁴ X 64 X 2²

=  2⁴ X 2⁶ X 2²

=  2⁴⁺⁶⁺²

=  2¹²        (Ans.)

2)  5⁴ X 5² X 625 X 125 = ?

Ans.)  5⁴ X 5² X 625 X 125

=  5⁴ X 5² X 5⁴ X 5

=  5⁴⁺²⁺⁴⁺

=   5¹ᶟ     (Ans.)

Express each of the following as a power of a number –

1)   (3⁴)² ÷ 3⁴ = ?

Ans.)   (3⁴)² ÷ 3⁴

=  3⁴ˣ² ÷ 3⁴

=  3⁸ ÷ 3⁴

=   3⁸⁻⁴

=  3⁴           (Ans.)

2)  (10⁵)⁴ ÷ (10⁴)ᶟ  = ?

Ans.)  (10⁵)⁴ ÷ (10⁴)ᶟ

=   10⁵ˣ⁴ ÷ 10⁴ˣ

=   10²⁰ ÷ 10¹²

=   10²⁰⁻¹²

=    10⁸     (Ans.)

Express each of the following as a power of a number –

1)  Express (64 X 625 )⁴ as a product of the powers of 4 & 5

Ans.)   (64 X 625 )⁴

=  ( 4 X 5⁴ )⁴

=  4ˣ⁴ X 5⁴ˣ

=   4¹² X 5¹⁶      (Ans.)

Express each of the following as a power of a number with a positive index –

1)  (10⁻⁵)⁶ = ?

Ans.)   (10⁻⁵)⁶

( 1 ) ⁶

=    -------------

(10⁵ )⁶

1

=    ------------

10⁵ˣ⁶

1

=   ------------         (Ans.)

10

2)  Find  ( 3xᶟy² )⁵ - (4x²y⁶)² = ?

Ans.)   ( 3xᶟy² )⁵ - (4x²y⁶)²

= 3⁵ xᶟˣ⁵ . y²ˣ⁵ - 4² x²ˣ² y⁶ˣ²

= 243 x⁸y⁷ - 16 x⁴y¹²           (Ans.)

( 4a⁴b⁻⁶ + 3a⁻²b⁵) X 3 a⁷b⁴

3)     -------------------------------

( 3a⁻⁸b⁵ - 5a⁶b⁷) X 6 a⁻⁴b⁻⁶

( 4a⁴b⁻⁶ + 3a⁻²b⁵) X 3 a⁷b⁴

Ans.)   ---------------------------------

( 3a⁻⁸b⁵ - 5a⁶b⁷) X 6 a⁻⁴b⁻⁶

(4a⁴b⁻⁶ X  3 a⁷b⁴)  + (3a⁻²b⁵ X 3 a⁷b⁴)

=  -----------------------------------------

(3a⁻⁸b⁵ X  6 a⁻⁴b⁻⁶) – (5a⁶b⁷ X 6 a⁻⁴b⁻⁶)

{(4X3) X (a⁴ X a⁷) X (b⁻⁶X b⁴)} + { (3 X 3) X (a⁻²X a⁷) X (b⁵ X b⁴)}

= --------------------------------------------------------

{(3X6) X (a⁻⁸ X a⁻⁴) X (b⁵ X b⁻⁶)} – {(5 6) X (a⁶ Xa⁻⁴) X (b⁷X b⁻⁶)}

(12 a⁴⁺⁷ b⁻⁶⁺⁴) + (9 a⁻²⁺⁷ b⁵⁺⁴)

=  ----------------------------------

(18 a⁻⁸⁻⁴ b⁵⁻⁶) – ( 30 a⁶⁻⁴b⁷⁻⁶)

12 a¹¹ b⁻² + 9 a⁵b⁹

=  ---------------------------               (Ans.)

18 a⁻¹² b⁻¹ - 30 a²b¹

( 4a⁴b⁻⁶ + 3a⁻²b⁵) X 3 a⁷b⁴

4)   ------------------------------

6 a⁻⁴b⁶

( 4a⁴b⁻⁶ + 3a⁻²b⁵) X 3 a⁷b⁴

Ans.)  -----------------------------------

6 a⁻⁴b⁶

(4a⁴b⁻⁶ X 3 a⁷b⁴) + (3a⁻²b⁵ X 3 a⁷b⁴)

=  -------------------------------------

6 a⁻⁴b⁶

{(4X3) X (a⁴ X a⁷) X (b⁻⁶ X b⁴)} + {(3X3) X (a⁻²X a⁷) X (b⁵ X b⁴)}

= ------------------------------------------------------

6 a⁻⁴b⁶

12 a⁴⁺⁷ b⁻⁶⁺⁴ + 9 a⁻²⁺⁷ b⁵⁺⁴

= -------------------------------

6 a⁻⁴b⁶

12 a¹¹ b⁻² + 9 a⁵ b⁹

= ----------------------------

6 a⁻⁴b⁶

12 a¹¹ b⁻²             9 a⁵ b⁹

=    ---------------  +  -------------

6 a⁻⁴b⁶                6 a⁻⁴b⁶

12                                     9

= -------  (a¹¹ b⁻²) X (a⁻⁴b⁶)⁻¹ +  --------  (a⁵ b⁹) X (a⁻⁴b⁶)⁻¹

6                                      6

=  2 (a¹¹ b⁻²) X (a⁻⁴ˣ⁻¹ b⁶ˣ ⁻¹) + 3/2 (a⁵ b⁹) X (a⁻⁴ˣ⁻¹ b⁶ˣ⁻¹)

=  2 (a¹¹ b⁻²) X (a⁴ b⁻⁶) + 3/2 (a⁵ b⁹) X (a⁴ b⁻⁶)

=  2 (a¹¹ X a⁴) X (b⁻² X b⁻⁶) + 3/2 (a⁵ X a⁴) X (b⁹ X b⁻⁶)

=  2 a¹¹⁺⁴ b⁻²⁻⁶ + 3/2 a⁵⁺⁴ b⁹⁻⁶

=  2 a¹⁵ b⁻⁸ + 3/2 a⁹ bᶟ           (Ans.)