$21^x \times 21^{6.5} = 21^{12.4}$

### Concept & Formula This problem tests the foundational product rule of indices. When multiplying two exponential terms with the same base, you simply add their exponents. Once the bases on both sides of an equation are identical, their exponents can be set equal to each other. $$a^m \times a^n = a^{m+n}$$ ### Step-by-Step Solution * **Given:** $$21^x \times 21^{6.5} = 21^{12.4}$$ * **Calculation / Deduction:** 1. Apply the product rule of exponents to the Left Hand Side (LHS) by combining the common base of 21: $$21^{x + 6.5} = 21^{12.4}$$ 2. Since the bases on both sides of the equation are equal (21) and non-zero, their corresponding exponents must be strictly equal: $$x + 6.5 = 12.4$$ 3. Solve for the unknown variable $x$ by subtracting 6.5 from 12.4: $$x = 12.4 - 6.5$$ $$x = 5.9$$ ### Exam Strategy & Shortcut **Mental Arithmetic:** Recognize immediately that the base 21 is irrelevant to the actual calculation. The problem instantly simplifies in your head to $x + 6.5 = 12.4$. Subtraction of simple decimals ($12.4 - 6.5$) yields $5.9$ directly. ### Common Pitfall Students rushing through the exam might mistakenly divide $12.4$ by $6.5$, confusing the exponent rules for multiplication with the rules for power of a power. Always remember that multiplication of bases means addition of exponents. ### Final Answer **Therefore, the correct answer is $5.9$.**