Expand
(i) `(3x+2)^3`
(ii) `(3a+(1)/(4b))^3`
(iii) ` (1+(2)/(3)a)^3`

Answer

To expand the expressions given in the question, we will use the formula for the cube of a binomial, which is:

\[ (a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2 \]

Now, let's solve each part step by step.

### (i) Expand \((3x + 2)^3\)

1. **Identify \(a\) and \(b\)**:
- Here, \(a = 3x\) and \(b = 2\).

2. **Calculate \(a^3\)**:
\[ (3x)^3 = 27x^3 \]

3. **Calculate \(b^3\)**:
\[ 2^3 = 8 \]

4. **Calculate \(3a^2b\)**:
\[ 3(3x)^2(2) = 3(9x^2)(2) = 54x^2 \]

5. **Calculate \(3ab^2\)**:
\[ 3(3x)(2^2) = 3(3x)(4) = 36x \]

6. **Combine all the terms**:
\[ (3x + 2)^3 = 27x^3 + 54x^2 + 36x + 8 \]

### Final Answer for (i):
\[ (3x + 2)^3 = 27x^3 + 54x^2 + 36x + 8 \]

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### (ii) Expand \((3a + \frac{1}{4b})^3\)

1. **Identify \(a\) and \(b\)**:
- Here, \(a = 3a\) and \(b = \frac{1}{4b}\).

2. **Calculate \(a^3\)**:
\[ (3a)^3 = 27a^3 \]

3. **Calculate \(b^3\)**:
\[ \left(\frac{1}{4b}\right)^3 = \frac{1}{64b^3} \]

4. **Calculate \(3a^2b\)**:
\[ 3(3a)^2\left(\frac{1}{4b}\right) = 3(9a^2)\left(\frac{1}{4b}\right) = \frac{27a^2}{4b} \]

5. **Calculate \(3ab^2\)**:
\[ 3(3a)\left(\frac{1}{4b}\right)^2 = 3(3a)\left(\frac{1}{16b^2}\right) = \frac{9a}{16b^2} \]

6. **Combine all the terms**:
\[ (3a + \frac{1}{4b})^3 = 27a^3 + \frac{1}{64b^3} + \frac{27a^2}{4b} + \frac{9a}{16b^2} \]

### Final Answer for (ii):
\[ (3a + \frac{1}{4b})^3 = 27a^3 + \frac{1}{64b^3} + \frac{27a^2}{4b} + \frac{9a}{16b^2} \]

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### (iii) Expand \((1 + \frac{2}{3}a)^3\)

1. **Identify \(a\) and \(b\)**:
- Here, \(a = 1\) and \(b = \frac{2}{3}a\).

2. **Calculate \(a^3\)**:
\[ 1^3 = 1 \]

3. **Calculate \(b^3\)**:
\[ \left(\frac{2}{3}a\right)^3 = \frac{8}{27}a^3 \]

4. **Calculate \(3a^2b\)**:
\[ 3(1^2)\left(\frac{2}{3}a\right) = 3(1)\left(\frac{2}{3}a\right) = 2a \]

5. **Calculate \(3ab^2\)**:
\[ 3(1)\left(\frac{2}{3}a\right)^2 = 3(1)\left(\frac{4}{9}a^2\right) = \frac{4}{3}a^2 \]

6. **Combine all the terms**:
\[ (1 + \frac{2}{3}a)^3 = 1 + \frac{8}{27}a^3 + 2a + \frac{4}{3}a^2 \]

### Final Answer for (iii):
\[ (1 + \frac{2}{3}a)^3 = 1 + \frac{8}{27}a^3 + 2a + \frac{4}{3}a^2 \]

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