Solve the exponential equation. Express irrational solutions in exact form. \[ 8^{x-7}=64 \]

To solve the equation \( 8^{x-7} = 64 \), we can express both sides with the same base. We can rewrite \( 8 \) and \( 64 \) as powers of \( 2 \): \[ 8 = 2^3 \quad \text{and} \quad 64 = 2^6. \] Now, substituting these values into the equation gives us: \[ (2^3)^{x-7} = 2^6. \] Applying the power of a power property, we have: \[ 2^{3(x-7)} = 2^6. \] Since the bases are the same, we can equate the exponents: \[ 3(x-7) = 6. \] Now, solving for \( x \): \[ 3x - 21 = 6 \\ 3x = 6 + 21 \\ 3x = 27 \\ x = \frac{27}{3} \\ x = 9. \] Thus, the solution to the equation is: \[ \boxed{9}. \]