Answer :
(i) a⁴ - b⁴ = (a2)2 - (b2)2
Using identity a2 - b2 = (a + b)(a - b)
Here a = a2 ; b = b2
(a2)2 - (b2)2 = (a2 + b2) (a2 - b2)
Again Using identity a2 - b2 = (a + b)(a - b)
Here a = a ; b = b
a2 - b2 = (a + b)(a - b)
(a2)2 - (b2)2 = (a2 + b2) (a + b)(a - b)
(ii) p⁴ - 81 = (p2)2 - (32)2
Using identity a2 - b2 = (a + b)(a - b)
Here a = p2 ; b = 32
(p2)2 - (32)2 = (p2 + 32) (p2 - 32)
Again Using identity a2 - b2 = (a + b)(a - b)
Here a = p ; b = 3
p2 - 32 = (p + 3)(p - 3)
(p2)2 - (32)2 = (p2 + 32) (p + 3)(p - 3)
(iii) x⁴ - (y + z)⁴ = (x2)2 - {(y + z)2}2
Using identity a2 - b2 = (a + b)(a - b)
Here a = x2 ; b = (y + z)2
(x2)2 - (y + z2)2 = {x2 + (y + z)2} {x2 - (y + z)2}
Again Using identity a2 - b2 = (a + b)(a - b)
Here a = x ; b = y + z
x2 - (y + z)2 = {x + (y + z)}{(x – (y + z)}
(x2)2 - (y + z2)2 = {x2 + (y + z)2} (x + y + z)(x – y - z)
(iv) x⁴ - (x - z)⁴ = (x2)2 - {x - z)2}2
Using identity a2 - b2 = (a + b)(a - b)
Here a = x2 ; b = (x - z)2
(x2)2 - (x - z2)2 = {x2 + (x - z)2} {x2 - (x - z)2}
Again Using identity a2 - b2 = (a + b)(a - b)
Here a = x ; b = x - z
x2 - (x - z)2 = {x + (x - z)}{(x – (x - z)}
(x2)2 - (x - z2)2 = {x2 + (x - z)2} (x + x - z)(x – x + z)
(x2)2 - (x - z2)2 = {x2 + (x - z)2} (2x - z)(z)
(x2)2 - (x - z2)2 = (2x2 -2xz + z2) (2x - z)(z)
[using (a + b)2 = a2 + b2 + 2ab]
(v) a⁴ - 2a2b2 + b⁴ = (a2)2 -2 × a× b + (b2)2
= (a2)2 -2 × a× b + (b2)2
[using (a - b)2 = a2 -2ab + b2]
= (a2 - b2)2
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