Answer :

(i) a⁴ - b⁴ = (a^{2})^{2} - (b^{2})^{2}

Using identity a^{2} - b^{2} = (a + b)(a - b)

Here a = a^{2} ; b = b^{2}

(a^{2})^{2} - (b^{2})^{2} = (a^{2} + b^{2}) (a^{2} - b^{2})

Again Using identity a^{2} - b^{2} = (a + b)(a - b)

Here a = a ; b = b

a^{2} - b^{2} = (a + b)(a - b)

(a^{2})^{2} - (b^{2})^{2} = (a^{2} + b^{2}) (a + b)(a - b)

(ii) p⁴ - 81 = (p^{2})^{2} - (3^{2})^{2}

Using identity a^{2} - b^{2} = (a + b)(a - b)

Here a = p^{2} ; b = 3^{2}

(p^{2})^{2} - (3^{2})^{2} = (p^{2} + 3^{2}) (p^{2} - 3^{2})

Again Using identity a^{2} - b^{2} = (a + b)(a - b)

Here a = p ; b = 3

p^{2} - 3^{2} = (p + 3)(p - 3)

(p^{2})^{2} - (3^{2})^{2} = (p^{2} + 3^{2}) (p + 3)(p - 3)

(iii) x⁴ - (y + z)⁴ = (x^{2})^{2} - {(y + z)^{2}}^{2}

Using identity a^{2} - b^{2} = (a + b)(a - b)

Here a = x^{2} ; b = (y + z)^{2}

(x^{2})^{2} - (y + z^{2})^{2} = {x^{2} + (y + z)^{2}} {x^{2} - (y + z)^{2}}

Again Using identity a^{2} - b^{2} = (a + b)(a - b)

Here a = x ; b = y + z

x^{2} - (y + z)^{2} = {x + (y + z)}{(x – (y + z)}

(x^{2})^{2} - (y + z^{2})^{2} = {x^{2} + (y + z)^{2}} (x + y + z)(x – y - z)

(iv) x⁴ - (x - z)⁴ = (x^{2})^{2} - {x - z)^{2}}^{2}

Using identity a^{2} - b^{2} = (a + b)(a - b)

Here a = x^{2} ; b = (x - z)^{2}

(x^{2})^{2} - (x - z^{2})^{2} = {x^{2} + (x - z)^{2}} {x^{2} - (x - z)^{2}}

Again Using identity a^{2} - b^{2} = (a + b)(a - b)

Here a = x ; b = x - z

x^{2} - (x - z)^{2} = {x + (x - z)}{(x – (x - z)}

(x^{2})^{2} - (x - z^{2})^{2} = {x^{2} + (x - z)^{2}} (x + x - z)(x – x + z)

(x^{2})^{2} - (x - z^{2})^{2} = {x^{2} + (x - z)^{2}} (2x - z)(z)

(x^{2})^{2} - (x - z^{2})^{2} = (2x^{2} -2xz + z^{2}) (2x - z)(z)

[using (a + b)^{2} = a^{2} + b^{2} + 2ab]

(v) a⁴ - 2a^{2}b^{2} + b⁴ = (a^{2})^{2} -2 × a× b + (b^{2})^{2}

= (a^{2})^{2} -2 × a× b + (b^{2})^{2}

[using (a - b)^{2} = a^{2} -2ab + b^{2}]

= (a^{2} - b^{2})^{2}

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