The sum of the coefficients in the expansion of `(x^(2)-1/3)^(199)(x^(3)+1/2)^(200)` is ______

Answer

To find the sum of the coefficients in the expansion of \((x^{2} - \frac{1}{3})^{199}(x^{3} + \frac{1}{2})^{200}\), we can follow these steps:

### Step 1: Understand the sum of coefficients
The sum of the coefficients in a polynomial can be found by substituting \(x = 1\) into the polynomial. This is because substituting \(x = 1\) effectively adds up all the coefficients.

### Step 2: Substitute \(x = 1\)
We substitute \(x = 1\) into the expression:
\[ (1^{2} - \frac{1}{3})^{199}(1^{3} + \frac{1}{2})^{200} \]

### Step 3: Simplify the expression
Now, we simplify each part:
\[ (1 - \frac{1}{3})^{199} = \left(\frac{2}{3}\right)^{199} \]
and
\[ (1 + \frac{1}{2})^{200} = \left(\frac{3}{2}\right)^{200} \]

### Step 4: Combine the results
Now, we combine the results:
\[ \left(\frac{2}{3}\right)^{199} \cdot \left(\frac{3}{2}\right)^{200} \]

### Step 5: Simplify the combined expression
We can simplify this expression:
\[ \left(\frac{2^{199}}{3^{199}}\right) \cdot \left(\frac{3^{200}}{2^{200}}\right) = \frac{2^{199} \cdot 3^{200}}{3^{199} \cdot 2^{200}} = \frac{3^{200}}{3^{199}} \cdot \frac{2^{199}}{2^{200}} = \frac{3^{1}}{2^{1}} = \frac{3}{2} \]

### Step 6: Final result
Thus, the sum of the coefficients in the expansion is:
\[ \frac{3}{2} = 1.5 \]

### Final Answer
The sum of the coefficients in the expansion is \(1.5\).

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