Answer :

**(i)** 1000a^{3}+27b^{6}

= (10a)^{3} + (3b^{2})^{6}

Using x^{3} + y^{3} = (x = y)(x^{2} - xy + y^{2})

= (10a + 3b^{2})[(10a)^{2} - (10a)(3b^{2}) + (3b^{2})^{2}]

= (10a + 3b^{2})(100a^{2} - 30ab^{2} + 9b^{4})

**(ii)** 1-216 z^{3}

= 1 – (6z)^{3}

Using x^{3} – y^{3} = (x – y)(x^{2} + xy + y^{2})

= (1 – 6z)(1^{2} + 1(6z) + (6z)^{2})

= (1 – 6z)(1 + 6z + 36z^{2})

**(iii)** m^{4} - m

= m(m^{3} – 1)

= m(m^{3} – 1^{3})

Using x^{3} – y^{3} = (x – y)(x^{2} + xy + y^{2})

= m(m – 1)(m^{2} + m + 1)

**(iv)** 192a^{3}+3

= 3(64a^{3} + 1)

= 3[(4a)^{3} + 1^{3}]

Using x^{3} + y^{3} = (x + y)(x^{2} – xy + y^{2})

= 3(4a + 1)[(4a)^{2} – 4a + 1]

= 3(4a + 1)(16a^{2} – 4a + 1)

**(v)** 16a^{4}x^{3} + 54ay^{3}

= 2a(8a^{3}x^{3} + 27y^{3})

= 2a[(2ax)^{3} + (3y)^{3}]

Using x^{3} + y^{3} = (x + y)(x^{2} – xy + y^{2})

= 2a(2ax + 3y)[(2ax)^{2} – 2ax(3y) + (3y)^{2}]

= 2a(2ax + 3y)(4a^{2}x^{2} – 6axy + 9y^{2})

**(vi)** 729a^{3} b^{3} c^{3}-125

= (9abc)^{3} - 5^{3}

= (9abc – 5)[(9abc)^{2} + 9abc(5) + 5^{2}]

= (9abc – 5)(81a^{2}b^{2}c^{2} + 45abc + 25)

**(vii)**

Using x^{3} - y^{3} = (x - y)(x^{2} + xy + y^{2})

**(viii)**

Using x^{3} – y^{3} = (x – y)(x^{2} + xy + y^{2})

**(ix)**x^{3} + 3x^{2}y + 3xy^{2} + 2y^{3}

= x^{3} + y^{3} + 3x^{2}y + 3xy^{2} + y^{3}

= (x + y)^{3} + y^{3}

[∵ a^{3} + b^{3} + 3a^{2}b + 3ab^{2} = (a + b)^{3}]

= (x + y + y)[(x + y)^{2} - (x + y)y + y^{2}]

[Using x^{3} – y^{3} = (x + y)(x^{2} - xy + y^{2})]

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

= (x + 2y)(x^{2} – xy + y^{2})

**(x)** 1 + 9x + 27x^{2} + 28x^{3}

= 27x^{3} + 1 + 27x^{2} + 9x + x^{3}

= (3x)^{3} + 1^{3} + 3(3x)^{3}(1) + 3(3x)(1)^{2}

= (3x + 1)^{3} + x^{3}

[∵ a^{3} + b^{3} + 3a^{2}b + 3ab^{2} = (a + b)^{3}]

Now, Using x^{3} + y^{3} = (x + y)(x^{2} - xy + y^{2})

= (3x + 1 + x)[(3x + 1)^{2} - (3x + 1)x + x^{2}]

= (4x + 1)(9x^{2} + 6x + 1 - 3x^{2} - x + x^{2})

= (4x + 1)(7x^{2} + 5x + 1)

**(xi)** x^{3} – 9y^{3} – 3xy(x-y)

x^{3} – y^{3} – 3xy (x – y) – 8y^{3}

[∵ x^{3} – y^{3} = x^{3} – y^{3} – 3xy (x – y)]

= (x – y)^{3} – (2y)^{3}

= (x – y – (2y)) [(x – y)^{2} + (x – y) (2y) + (2y)^{2}]

[Using x^{3} + y^{3} = (x + y) (x^{2} – xy + y^{2})]

= (x -3 y) (x^{2} + y^{2} – 2xy + 2xy – 2y^{2} + 4y^{2})

= (x - 3y) (x^{2} + 3y^{2})

**(xii)**8 – a^{3} + 3a^{2}b – 3ab^{2} + b^{3}

= b^{3} – a^{3} – 3ab^{2} + 3a^{2}b + 8

Now, As (x + y)^{3} = x^{3} + y^{3} + 3x^{2}y + 3xy^{2}

= (b – a)^{3} + 2^{3}

Using x^{3} + y^{3} = (x + y) (x^{2} - xy + y^{2})

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

= (b – a + 2) (b^{2} + a^{2} – 2ab - 2b + 2a + 4)

**(xiii)** x^{6}+3x^{4} b^{2}+3x^{2} b^{4}+b^{6}+a^{3}b^{3}

_{= (x}^{2})^{3} + (b^{2})^{3} + 3(x^{2})^{2}(b^{2}) + 3(x^{2})(b^{2})^{2} + a^{3}b^{3}

_{As (x + y)}^{3} = x^{3} + y^{3} + 3x^{2}y + 3xy^{2}

_{= (x}^{2} + b^{2})^{3} + (ab)^{3}

_{Using x}^{3} + y^{3} = (x + y)(x^{2} – xy + y^{2})

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

**(xiv)** x^{6} + 27

= (x^{2})^{3} + (3)^{3}

Using x^{3} + y^{3} = (x + y)(x^{2} – xy + y^{2})

= (x^{2} + 3)[(x^{2})^{2} - 3x^{2} + 3^{2})

= (x^{2} + 3)(x^{4} - 3x^{2} + 9)

**(xv)**x^{6} – y^{6}

= (x^{3})^{2} – (y^{3})^{2}

Using (a^{2} – b^{2}) = (a – b)(a + b)

= (x^{3} – y^{3})(x^{3} + y^{3})

Using x^{3} – y^{3} = (x – y)(x^{2} + xy + y^{2}) and

x^{3} + y^{3} = (x + y)(x^{2} – xy + y^{2})

= (x – y)(x^{2} + xy + y^{2})(x + y)(x^{2} – xy + y^{2})

**(xvi)** x^{12} – y^{12}

= (x^{6})^{2} – (y^{6})^{2}

Using (a^{2} – b^{2}) = (a – b)(a + b)

= (x^{6} – y^{6})(x^{6} + y^{6})

= [(x^{3})^{2} – (y^{3})^{2 �}][(x^{2})^{3} + (y^{2})^{3}]

Using (a^{2} – b^{2}) = (a – b)(a + b)

= (x^{3} – y^{3})(x^{3} + y^{3})[(x^{2})^{3} + (y^{2})^{3}]

Using x^{3} – y^{3} = (x – y)(x^{2} + xy + y^{2}) and

x^{3} + y^{3} = (x + y)(x^{2} – xy + y^{2})

= (x – y)(x^{2} + xy + y^{2})(x + y)(x^{2} – xy + y^{2})(x^{2} + y^{2})(x^{4} - x^{2}y^{2} + x^{6}y^{6})

**(xvii)** m^{3} -n^{3}-m(m^{2}-n^{2} )+n(m-n)^{2}

Using x^{3} – y^{3} = (x – y)(x^{2} + xy + y^{2}) and

Using (a^{2} – b^{2}) = (a – b)(a + b)

= (m – n)(m^{2} + mn + n^{2}) – m(m – n)(m + n) + n(m – n)^{2}

Taking (m – n) as common, we get

= (m – n)[m^{2} + mn + n^{2} – m(m + n) + n(m – n)]

= (m – n)(m^{2} + mn + n^{2} – m^{2} – mn + mn – n^{2})

= (m – n)mn

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