Express each of the following as a product of prime factors only in exponential form:

(i) 108 × 192

(ii) 270

(iii) 729 × 64

(iv) 768

(i) We have,

108 × 192

We have to express this as a product of prime factors only in exponential form

∴ 108 × 192

= (2 × 2 × 3 × 3 × 3) × (2 × 2 × 2 × 2 × 2 × 2 × 3)

= (2² × 3³) × (2⁶ × 3)

Using identity: (a^m × aⁿ = a^{m + n})

= 2^{6 + 2} × 3^{3 + 1}

= 2⁸ × 3⁴

(ii) We have,

270

We have to express this as a product of prime factors only in exponential form

Thus,

270

= 2 × 3 × 3 × 3 × 5

= 2 × 3³ × 5

(iii) We have,

729 × 64

We have to express this as a product of prime factors only in exponential form

Thus,

729 × 64

= (3 × 3 × 3 × 3 × 3 × 3) × (2 × 2 × 2 × 2 × 2 × 2)

= 3⁶ × 2⁶

(iv) We have,

768

We have to express this as a product of prime factors only in exponential form

Thus, 768 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3

= 2⁸ × 3