$(25)^{7.5} \times (5)^{2.5} \div (125)^{1.5} = 5^{x}$

### Concept & Formula This problem requires converting various composite bases into a common prime base to simplify the expression using the **Laws of Indices**. The key formulas are: $$ (a^m)^n = a^{m \times n} $$ $$ a^m \times a^n = a^{m+n} $$ $$ a^m \div a^n = a^{m-n} $$ ### Step-by-Step Solution * **Step 1:** Identify the target base from the Right Hand Side, which is 5. Convert 25 and 125 to powers of 5. $25^{7.5} = (5^2)^{7.5} = 5^{15}$ $125^{1.5} = (5^3)^{1.5} = 5^{4.5}$ * **Step 2:** Substitute these simplified terms back into the original equation. $5^{15} \times 5^{2.5} \div 5^{4.5} = 5^x$ * **Step 3:** Apply the laws of indices for multiplication (addition of exponents) and division (subtraction of exponents). $5^{15 + 2.5 - 4.5} = 5^x$ * **Step 4:** Simplify the exponent. $15 + 2.5 - 4.5 = 13$ $5^{13} = 5^x$ * **Step 5:** Equate the exponents. $x = 13$ ### Exam Strategy & Shortcut Skip writing out the base conversions. Map each given base directly to its power of 5 (25 is $5^2$, 125 is $5^3$). Multiply these powers by the given exponents mentally: $2 \times 7.5 = 15$ and $3 \times 1.5 = 4.5$. Then, just write the final exponent arithmetic: $15 + 2.5 - 4.5 = 13$. ### Common Pitfall Students sometimes add all exponents blindly or misplace the decimal point during multiplication (e.g., $3 \times 1.5 = 45$ instead of 4.5), completely throwing off the final result. ### Final Answer **Therefore, the correct answer is 13.**