To simplify the expression \(\left(\frac{3}{2}x + 1\right)^3\), follow these steps:

1. **Identify the expression inside the parentheses**:
   \[
   \frac{3}{2}x + 1
   \]
   This expression represents a linear term where \(x\) is a variable.

2. **Raise the entire expression to the power of 3**:
   \[
   \left(\frac{3}{2}x + 1\right)^3
   \]

   This can be expanded using the binomial theorem, which states that \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\).

   Here, \(a = \frac{3}{2}x\) and \(b = 1\).

3. **Expand the expression**:
   \[
   \left(\frac{3}{2}x\right)^3 + 3\left(\frac{3}{2}x\right)^2(1) + 3\left(\frac{3}{2}x\right)(1^2) + 1^3
   \]

   Now, calculate each term:
   - \(\left(\frac{3}{2}x\right)^3 = \frac{27}{8}x^3\)
   - \(3\left(\frac{3}{2}x\right)^2(1) = 3 \times \frac{9}{4}x^2 = \frac{27}{4}x^2\)
   - \(3\left(\frac{3}{2}x\right)(1^2) = 3 \times \frac{3}{2}x = \frac{9}{2}x\)
   - \(1^3 = 1\)

   Combine all the terms:
   \[
   \frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1
   \]

So, \(\left(\frac{3}{2}x + 1\right)^3\) expands to \(\frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1\).

Would you like more details on the expansion, or do you have any questions?

Here are some related questions you might be interested in:
1. How do you expand \((2x + 3)^3\) using the binomial theorem?
2. What is the cube of \(\frac{2}{3}x + 1\)?
3. How do you simplify \(\left(\frac{3}{2}x - 1\right)^3\)?
4. How do you find the derivative of \(\left(\frac{3}{2}x + 1\right)^3\)?
5. What is the expansion of \((ax + b)^3\) in general terms?

**Tip**: The binomial theorem is a powerful tool for expanding expressions raised to a power, especially when dealing with polynomials.

Learn how to expand the expression (3/2x + 1)^3 using the Binomial Theorem with a detailed step-by-step solution. This problem covers polynomial expansion techniques and the application of the Binomial Theorem formula. Suitable for students in Grades 10-12.

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