Let’s fill up the table below :

(b)

First Part,

Given, (i) x⁴ – 4x³ + 6x²

(ii) x²

(i) × (ii)

= (x⁴ – 4x³ + 6x²) × x²

= x⁴x² – 4x³x² + 6x²x²

We know (x^ax^b= x^a+b)

= x⁴⁺²– 4x³⁺² + 6x²⁺²

= x⁶ – 4x⁵ + 6x⁴

(i) ÷ (ii)

We know

= x^4-2 – 4x^3-2 + 6x^2-2

= x² – 4x + 6

Second Part

Given, (i) 3m²n³ + 40m³n⁴ – 5m⁴n⁵

(ii) 10m²n²

(i) × (ii)

= (3m²n³ + 40m³n⁴ – 5m⁴n⁵) × 10m²n²

We know (x^ax^b= x^a+b)

= 30m²m²n³n² + 400m³m²n⁴n² – 50m⁴m²n⁵n²

= 30m²⁺²n³⁺² + 400m³⁺²n⁴⁺² – 50m⁴⁺²n⁵⁺²

= 30m⁴n⁵ + 400m⁵n⁶ – 50m⁶n⁷

(i) ÷ (ii)

We know

(c)

Given,

(i) 49l² – 100m²

(ii) (7l + 10m)

(i) × (ii)

= (49l² – 100m²)(7l + 10m)

We know (x^ax^b= x^a+b)

= 343l²l+ 490l²m – 700m²l – 1000m²m

= 343l²⁺¹ + 490l²m – 700lm² – 1000m²⁺¹

= 343l³ + 490l²m – 700lm² – 1000m³

(i) ÷ (ii)

∵ a² – b² = (a – b)(a + b), taking a = 7l, b = 10m we have

Cancelling out the common terms from numerator and denominator, we get

= 7l – 10m

(d)

Given,

(i) 625a⁴ – 81b⁴

(ii) 5a + 3b

(i) × (ii)

= (625a⁴ – 81b⁴)(5a + 3b)

= 625a⁴a + 1875a⁴b – 405ab⁴ – 243b⁴b

We know (x^ax^b= x^a+b)

= 625a⁴⁺¹ + 1875a⁴b – 405ab⁴ – 243b⁴⁺¹

= 3125a⁵ + 1875a⁴b – 405ab⁴ – 243b⁵

(i) ÷ (ii)

∵ a² – b² = (a – b)(a + b), taking a = 25a², b = 9b², we have

∵ a² – b² = (a – b)(a + b), taking a = 5a, b = 3b we have

=

Cancelling out the common terms from numerator and denominator, we get

= (5a – 3b)(25a² + 9b²)

= 125aa² + 45ab² – 75a²b – 27bb²

We know (x^ax^b= x^a+b)

= 125a²⁺¹ + 45ab² – 75a²b – 27b²⁺¹

= 125a³ + 45ab² – 75a²b – 27b³