Suppose the sum of the coefficients in the expansion of `(1 - 5x + 12x^(3))^(n)` is a and the sum of the coefficients in the expansion of `(1 - 2x + 3x^(2))^n` is b. If `a = b^(m)`, then m = ___

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) based on the given expressions, and then determine the relationship between them.

### Step 1: Find the sum of the coefficients for \( (1 - 5x + 12x^3)^n \)

To find the sum of the coefficients in the expansion of \( (1 - 5x + 12x^3)^n \), we substitute \( x = 1 \):

\[ a = (1 - 5(1) + 12(1)^3)^n = (1 - 5 + 12)^n = (8)^n \]

### Step 2: Find the sum of the coefficients for \( (1 - 2x + 3x^2)^n \)

Similarly, for the expression \( (1 - 2x + 3x^2)^n \), we substitute \( x = 1 \):

\[ b = (1 - 2(1) + 3(1)^2)^n = (1 - 2 + 3)^n = (2)^n \]

### Step 3: Relate \( a \) and \( b \)

We have found:

\[ a = 8^n \quad \text{and} \quad b = 2^n \]

Now, we need to express \( a \) in terms of \( b \):

\[ a = 8^n = (2^3)^n = (2^n)^3 = b^3 \]

### Step 4: Determine \( m \)

From the relationship \( a = b^m \), we can see that:

\[ a = b^3 \]

Thus, we can conclude that:

\[ m = 3 \]

### Final Answer

The value of \( m \) is \( \boxed{3} \).

---