To find \( m - n \), where \( m \) and \( n \) are natural numbers with \( n>1 \), and given the equation \( m^n = 2^{25} \times 3^{40} \), follow these steps:
- Represent \( m \) using its prime factors:
\( m = 2^a \times 3^b \) - Substitute this expression for \( m \) into the given equation:
\( (2^a \times 3^b)^n = 2^{25} \times 3^{40} \) - Apply the exponent rule \( (x^p)^q = x^{pq} \):
\( 2^{an} \times 3^{bn} = 2^{25} \times 3^{40} \) - Equate the exponents for each prime base:
\( an = 25 \),
\( bn = 40 \) - Solve for \( a \) and \( b \) in terms of \( n \):
\( a = \frac{25}{n} \),
\( b = \frac{40}{n} \) - Since \( a \) and \( b \) must be integers, \( n \) must be a common divisor of 25 and 40. The greatest common divisor (GCD) of 25 and 40 is 5. Since \( n>1 \), \( n \) must be 5.
Therefore, \( n = 5 \) - Substitute \( n = 5 \) back into the equations for \( a \) and \( b \):
\( a = \frac{25}{5} = 5 \),
\( b = \frac{40}{5} = 8 \) - Calculate the value of \( m \):
\( m = 2^5 \times 3^8 = 32 \times 6561 = 209952 \) - Calculate \( m - n \):
\( 209952 - 5 = \mathbf{209947} \)
The value of \( m - n \) is 209947.