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

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 \((a - b)^3 = a^3 - b^3 - 3ab(a - b)\)
\(= x^3 - 8y^3/27 - 2xy(x - 2y/3)\)
\(= x^3 - 8y^3/27 - 2x^2y + 4xy^2/3)\)
\(= x^3 - 2x^2y + 4xy^2/3 + - 8y^3/27\)