EXPRESS A NUMBER IN INDEX FORM

Express the following as the product of prime factors in index form:

Problem 1 :

56

Solution :

56 = 8 × 7

= 2 × 2 × 2 × 7

= 2³ × 7

Problem 2 :

240

Solution :

240 = 16 × 15

= 4 × 4 × 3 × 5

= 2 × 2 × 2 × 2 × 3 × 5

= 2⁴ × 3 × 5

Problem 3 :

504

Solution :

504 = 72 × 7

= 8 × 9 × 7

= 2 × 2 × 2 × 3 × 3 × 7

= 2³ × 3² × 7

Problem 4 :

735

Solution :

735 = 15 × 49

= 3 × 5 × 7 × 7

= 3 × 5 × 7²

Problem 5 :

297

Solution :

 297 = 27 × 11

= 9 × 3 × 11

= 3 × 3 × 3 × 11

= 3³ × 11

Problem 6 :

221

Solution :

221 = 13 × 17

Problem 7 :

360

Solution :

360 = 72 × 5

= 8 × 9 × 5

= 2 × 2 × 2 × 3 × 3 × 5

= 2³ × 3² × 5

Problem 8 :

952

Solution :

952 = 56 × 17

= 8 × 7 × 17

= 2 × 2 × 2 × 7 × 17

= 2³ × 7 × 17

Problem 9 :

Express each of the following products in index form

a) 2 x 2 x 2

b) 4 x 4

c) a x a x a x a

d) k x k x k

e) 10 x 10

f) 2 x 2 x 2 x 2 x 2 x 2

g) b x b x b x b x b

h) y x y x y x y x y x y x y x y

Solution :

a) 2 x 2 x 2

Since 2 is repeating three times, then the required index form is 2³.

b) 4 x 4

Since 4 is repeating two times, then the required index form is 4².

c) a x a x a x a

Since a is repeating four times, then the required index form is a⁴.

d) k x k x k

Since k is repeating three times, then the required index form is k³.

e) 10 x 10

Since 10 is repeating two times, then the required index form is 10².

f) 2 x 2 x 2 x 2 x 2 x 2

Since 2 is repeating six times, then the required index form is 2⁶.

g) b x b x b x b x b

Since b is repeating five times, then the required index form is b⁵.

h) y x y x y x y x y x y x y x y

Since y is repeating eight times, then the required index form is y⁸.

Problem 10 :

Write the following in expanded form

Solution :

a) 8² = 8 x 8

b) 3⁵ = 3 x 3 x 3 x 3 x 3

c) a⁴ = a x a x a x a

d) y³ = y x y x y

e) 5³ = 5 x 5 x 5

f) 9⁴ = 9 x 9 x 9 x 9

g) t⁶ = t x t x t x t x t x t

h) b⁵ = b x b x b x b x b

Problem 11 :

Evaluate the following

a) 2⁴

c) 3²

b) 5³

d) 1²⁰

Solution :

a) 2⁴ = 2 x 2 x 2 x 2

= 16

b) 3² = 3 x 3

= 9

c) 5³ = 5 x 5 x 5

= 125

d) 1²⁰

Multiplying 1 by any number of times, we get 1.

Problem 12 :

Write the prime factors of these numbers in index form

a) 20

c) 36

b) 96

d) 100

Solution :

a) 20 = 2 x 2 x 5

= 2² x 5

b) 36 = 3 x 3 x 2 x 2

= 3² x 2²

c) 96 = 3 x 2 x 2 x 2 x 2 x 2

= 3 x 2⁵

d) 100 = 5 x 5 x 2 x 2

= 5² x 2²

Problem 13 :

Write the following in index form

a) 2 x a x b x a x 2 x a x 2 x b

b) 4d x 2f x 4d x 2f

c) 2cd(2cd)(2cd)(2cd)

d) 3a x 2b x 3a x 2b

e) 5ab(5ab)5b

f) 7ab x 7ab x ab x 7

Solution :

a) 2 x a x b x a x 2 x a x 2 x b = 2³ x a³ x b²

b) 4d x 2f x 4d x 2f = (2f)² x (4d)²

c) 2cd(2cd)(2cd)(2cd) = (2cd)⁴

d) 3a x 2b x 3a x 2b = (3a)² x (2b)²

e) 5ab(5ab)5b = (5b)³ x a²

f) 7ab x 7ab x ab x 7 = 7³ x (ab)³