If a and b are positive ingegers, such that `a^b = 125, " then " (a - b)^(a+b-4) = ?`

Options

A. 16
B. 25
C. 28
D. 30

Answer

To solve the problem step by step, we start with the equation given:

1. **Identify the equation**: We know that \( a^b = 125 \).

2. **Express 125 in terms of powers**: We can express 125 as \( 5^3 \). Therefore, we have:
\[ a^b = 5^3 \]

3. **Set the bases equal**: Since both sides of the equation have the same base, we can equate the exponents:
\[ a = 5 \quad \text{and} \quad b = 3 \]

4. **Substitute values into the expression**: Now we need to calculate \( (a - b)^{(a + b - 4)} \). Substitute \( a = 5 \) and \( b = 3 \):
\[ (5 - 3)^{(5 + 3 - 4)} \]

5. **Simplify the expression**: Calculate \( 5 - 3 \) and \( 5 + 3 - 4 \):
\[ (2)^{(4)} \]

6. **Calculate the power**: Now, calculate \( 2^4 \):
\[ 2^4 = 16 \]

Thus, the final answer is:
\[ \boxed{16} \]