Step-by-step Solution:

We are given the equation: \[ (25)^{7.5} \times (5)^{2.5} \div (125)^{1.5} = 5^x \] Step 1: Express all numbers as powers of 5
We rewrite the given numbers in terms of base 5: \[ 25 = 5^2, \quad 125 = 5^3 \] Now, substitute these values into the equation: \[ (5^2)^{7.5} \times (5)^{2.5} \div (5^3)^{1.5} = 5^x \] Step 2: Apply the Power Rule
Using \((a^m)^n = a^{m \cdot n}\): \[ 5^{(2 \times 7.5)} \times 5^{2.5} \div 5^{(3 \times 1.5)} = 5^x \] \[ 5^{15} \times 5^{2.5} \div 5^{4.5} = 5^x \] Step 3: Simplify the Exponents
Using \( a^m \times a^n = a^{m+n} \): \[ 5^{(15 + 2.5)} \div 5^{4.5} = 5^x \] \[ 5^{17.5} \div 5^{4.5} = 5^x \] Using \( a^m \div a^n = a^{m-n} \): \[ 5^{(17.5 - 4.5)} = 5^x \] \[ 5^{13} = 5^x \] Thus, x = 13. Final Answer \( {13} \)