# 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 (xaxb= xa+b)

= x4+2– 4x3+2 + 6x2+2

= x6 – 4x5 + 6x4

(i) ÷ (ii)

We know

= x4-2 – 4x3-2 + 6x2-2

= x2 – 4x + 6

Second Part

Given, (i) 3m2n3 + 40m3n4 – 5m4n5

(ii) 10m2n2

(i) × (ii)

= (3m2n3 + 40m3n4 – 5m4n5) × 10m2n2

We know (xaxb= xa+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 (xaxb= xa+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 (xaxb= xa+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 (xaxb= xa+b)

= 125a2+1 + 45ab2 – 75a2b – 27b2+1

= 125a3 + 45ab2 – 75a2b – 27b3

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