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If for any two natural numbers a and b, ...

If for any two natural numbers a and b, `a^(b)=125` then `b^(a)` is

A

243

B

241

C

242

D

247

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the value of \( b^a \) given that \( a^b = 125 \). ### Step-by-Step Solution: 1. **Identify the expression**: We start with the equation \( a^b = 125 \). 2. **Express 125 in terms of powers**: We can express 125 as a power of 5. Specifically, \( 125 = 5^3 \). 3. **Set up the equation**: Therefore, we can rewrite the equation as: \[ a^b = 5^3 \] 4. **Determine possible values for \( a \) and \( b \)**: Since \( a \) and \( b \) are natural numbers, we can consider the following pairs: - If \( a = 5 \), then \( b = 3 \) because \( 5^3 = 125 \). - Alternatively, if \( a = 25 \), then \( b = 2 \) because \( 25^2 = 625 \) which is not valid. - The only valid pair is \( (5, 3) \). 5. **Calculate \( b^a \)**: Now we need to find \( b^a \) using the values we found: \[ b^a = 3^5 \] 6. **Compute \( 3^5 \)**: We can calculate \( 3^5 \) as follows: \[ 3^5 = 3 \times 3 \times 3 \times 3 \times 3 = 243 \] 7. **Final answer**: Thus, the value of \( b^a \) is: \[ b^a = 243 \] ### Summary: The final answer is \( b^a = 243 \).
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