What is the number of terms in the expansion of the following?
`(4x^(2) + 12 xy+ 9y^(2) )^(9)`

Answer

To find the number of terms in the expansion of \((4x^{2} + 12xy + 9y^{2})^{9}\), we can follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ (4x^{2} + 12xy + 9y^{2})^{9} \]

### Step 2: Recognize the form of the expression
Notice that \(4x^{2} + 12xy + 9y^{2}\) can be rewritten in a more recognizable form. It can be expressed as:
\[ (2x)^{2} + 2(2x)(3y) + (3y)^{2} \]
This is in the form of \(a^{2} + 2ab + b^{2}\), which is \((a + b)^{2}\).

### Step 3: Rewrite the expression
Thus, we can rewrite the original expression as:
\[ (2x + 3y)^{2} \text{ raised to the power of } 9 \]
This simplifies to:
\[ (2x + 3y)^{18} \]

### Step 4: Determine the number of terms
The number of terms in the expansion of \((a + b)^{n}\) is given by the formula \(n + 1\). In our case, \(n = 18\).

### Step 5: Calculate the number of terms
Using the formula:
\[ \text{Number of terms} = 18 + 1 = 19 \]

### Final Answer
Therefore, the number of terms in the expansion of \((4x^{2} + 12xy + 9y^{2})^{9}\) is:
\[ \boxed{19} \]