The number of terms in the expansion of `(x^(2)+6x+9)^(30)` is

Options

A. 31
B. 61
C. 91
D. none of these

Answer

To find the number of terms in the expansion of \((x^2 + 6x + 9)^{30}\), we can follow these steps:

### Step 1: Recognize the expression
The expression \(x^2 + 6x + 9\) can be rewritten using the perfect square formula. We know that:
\[ a^2 + 2ab + b^2 = (a + b)^2 \]
Here, we can identify \(a = x\) and \(b = 3\) since:
\[ x^2 + 6x + 9 = (x + 3)^2 \]

### Step 2: Rewrite the expression
Now we can rewrite the original expression:
\[ (x^2 + 6x + 9)^{30} = ((x + 3)^2)^{30} \]

### Step 3: Simplify the exponent
Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we simplify:
\[ ((x + 3)^2)^{30} = (x + 3)^{60} \]

### Step 4: Determine the number of terms
The number of terms in the expansion of \((x + y)^n\) is given by \(n + 1\). In our case, \(n = 60\):
\[ \text{Number of terms} = 60 + 1 = 61 \]

### Final Answer
Thus, the number of terms in the expansion of \((x^2 + 6x + 9)^{30}\) is \(\boxed{61}\).

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