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निम्नलिखित घनों को प्रसारित रूप में लिखिए। (i) (3/2 x + 1)3 (ii) (x/2 - y/3)3

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2 parts
i

278x3+274x2+92x+1\frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1

ii

x38−x2y4+xy26−y327\frac{x^3}{8} - \frac{x^2y}{4} + \frac{xy^2}{6} - \frac{y^3}{27}

Step-by-step solution

6 steps

Here are the expanded forms of the given expressions.

Part i· 3 steps

We need to expand (32x+1)3\left(\frac{3}{2}x + 1\right)^3.

We use the identity (a+b)3=a3+3a2b+3ab2+b3(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3.
Here, a=32xa = \frac{3}{2}x and b=1b = 1.

  1. Substitute aa and bb into the formula.

    (32x+1)3=(32x)3+3(32x)2(1)+3(32x)(1)2+(1)3\left(\frac{3}{2}x + 1\right)^3 = \left(\frac{3}{2}x\right)^3 + 3\left(\frac{3}{2}x\right)^2(1) + 3\left(\frac{3}{2}x\right)(1)^2 + (1)^3

  2. Simplify each term.

    =3323x3+3(3222x2)(1)+3(32x)(1)+1= \frac{3^3}{2^3}x^3 + 3\left(\frac{3^2}{2^2}x^2\right)(1) + 3\left(\frac{3}{2}x\right)(1) + 1 =278x3+3(94x2)+92x+1= \frac{27}{8}x^3 + 3\left(\frac{9}{4}x^2\right) + \frac{9}{2}x + 1

  3. Perform the multiplications.

    =278x3+274x2+92x+1= \frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1

    The expanded form is:

    278x3+274x2+92x+1\boxed{\frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1}

Part ii

x38−x2y4+xy26−y327\frac{x^3}{8} - \frac{x^2y}{4} + \frac{xy^2}{6} - \frac{y^3}{27}

We need to expand (x2−y3)3\left(\frac{x}{2} - \frac{y}{3}\right)^3.

We use the identity (a−b)3=a3−3a2b+3ab2−b3(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.
Here, a=x2a = \frac{x}{2} and b=y3b = \frac{y}{3}.

  1. Substitute aa and bb into the formula.

    (x2−y3)3=(x2)3−3(x2)2(y3)+3(x2)(y3)2−(y3)3\left(\frac{x}{2} - \frac{y}{3}\right)^3 = \left(\frac{x}{2}\right)^3 - 3\left(\frac{x}{2}\right)^2\left(\frac{y}{3}\right) + 3\left(\frac{x}{2}\right)\left(\frac{y}{3}\right)^2 - \left(\frac{y}{3}\right)^3

  2. Simplify each term.

    =x323−3(x222)(y3)+3(x2)(y232)−y333= \frac{x^3}{2^3} - 3\left(\frac{x^2}{2^2}\right)\left(\frac{y}{3}\right) + 3\left(\frac{x}{2}\right)\left(\frac{y^2}{3^2}\right) - \frac{y^3}{3^3} =x38−3(x24)(y3)+3(x2)(y29)−y327= \frac{x^3}{8} - 3\left(\frac{x^2}{4}\right)\left(\frac{y}{3}\right) + 3\left(\frac{x}{2}\right)\left(\frac{y^2}{9}\right) - \frac{y^3}{27}

  3. Perform the multiplications and simplify the fractions.

    =x38−3x2y12+3xy218−y327= \frac{x^3}{8} - \frac{3x^2y}{12} + \frac{3xy^2}{18} - \frac{y^3}{27} =x38−x2y4+xy26−y327= \frac{x^3}{8} - \frac{x^2y}{4} + \frac{xy^2}{6} - \frac{y^3}{27}

    The expanded form is:

    x38−x2y4+xy26−y327\boxed{\frac{x^3}{8} - \frac{x^2y}{4} + \frac{xy^2}{6} - \frac{y^3}{27}}
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