6.Write the following cubes in expanded form.
i)$$(2x + 1)^3$$
ii)$$(2a - 3b)^3$$
iii)$$[3x/2 + 1]^3$$
iv)$$(x - 2y/3)^3$$

i)$$(2x + 1)^3 = (2x)^3 + (1)^3 + 3(2x)(1)(2x + 1)$$
by using identity$$(a + b)^3 = a^3 + b^3 + 3ab(a + b)$$
$$= 8x^3 + 1 + 6x(2x + 1))$$
$$= 8x^3 + 1 + 12x^2 + 6x$$
$$= 8x^3 + 12x^2 + 6x + 1$$

ii)$$(2a - 3b)^3 = (2a)^3 - (3b)^3 - 3(2a)(3b)(2a - 3b)$$
by using identity $$(a - b)^3 = a^3 - b^3 - 3ab(a - b)$$
$$= 8a^3 - 27b^3 - 18ab(2a - 3b))$$
$$= 8a^3 - 27b^3 - 36a^2b + 54ab^2)$$
$$= 8a^3 - 36a^2b + 54ab^2 - 27b^3$$

iii)$$[3x/2 + 1]^3 = (3x/2)^3 + (1)^3 + 3(3x/2)(1)(3x/2 + 1)$$
by using identity $$(a + b)^3 = a^3 + b^3 + 3ab(a + b)$$
$$= 27x^3/8 + 1 + 9x/2(3x/2 + 1))$$
$$= 27x^3/8 + 1 + 27x^2/4 + 9x/2$$
$$= 27x^3/8 + 27x^2/4 + 9x/2 + 1$$

iv)$$(x - 2y/3)^3 = (x)^3 - (2y/3)^3 - 3(x)(2y/3)(x - 2y/3)$$
by using identity $$(a - b)^3 = a^3 - b^3 - 3ab(a - b)$$
$$= x^3 - 8y^3/27 - 2xy(x - 2y/3)$$
$$= x^3 - 8y^3/27 - 2x^2y + 4xy^2/3)$$
$$= x^3 - 2x^2y + 4xy^2/3 + - 8y^3/27$$