Expand : (i) `(3x + 2)^3`

Answer

To expand the expression \((3x + 2)^3\), we can use the formula for the cube of a binomial, which is given by:

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

Here, we can identify \(a = 3x\) and \(b = 2\).

### Step 1: Identify \(a\) and \(b\)
- Let \(a = 3x\) and \(b = 2\).

### Step 2: Calculate \(a^3\) and \(b^3\)
- Calculate \(a^3 = (3x)^3 = 27x^3\).
- Calculate \(b^3 = 2^3 = 8\).

### Step 3: Calculate \(3ab\)
- Calculate \(3ab = 3 \cdot (3x) \cdot 2 = 18x\).

### Step 4: Substitute into the formula
Now substitute \(a^3\), \(b^3\), and \(3ab\) into the formula:

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

This becomes:

\[ (3x + 2)^3 = 27x^3 + 8 + 18x(3x + 2) \]

### Step 5: Expand \(18x(3x + 2)\)
Now, we need to expand \(18x(3x + 2)\):

\[ 18x(3x + 2) = 54x^2 + 36x \]

### Step 6: Combine all parts
Now, combine all the parts together:

\[ (3x + 2)^3 = 27x^3 + 8 + 54x^2 + 36x \]

### Final Answer
Thus, the expanded form of \((3x + 2)^3\) is:

\[ 27x^3 + 54x^2 + 36x + 8 \]

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