$8^{2.4} \times 2^{3.7} \div (16)^{1.3} = 2^{x}$

### Concept & Formula This problem requires applying the **Laws of Indices** to simplify an expression by finding a common prime base. The essential exponent rules are: $$ (a^m)^n = a^{m \times n} $$ $$ a^m \times a^n \div a^p = a^{m+n-p} $$ ### Step-by-Step Solution * **Step 1:** Identify the lowest common base, which is 2. Convert all other bases (8 and 16) to powers of 2. $8^{2.4} = (2^3)^{2.4} = 2^{7.2}$ $(16)^{1.3} = (2^4)^{1.3} = 2^{5.2}$ * **Step 2:** Substitute these converted values back into the original equation. $2^{7.2} \times 2^{3.7} \div 2^{5.2} = 2^x$ * **Step 3:** Apply the multiplicative and divisional laws of indices to combine the exponents on the Left Hand Side (LHS). $2^{7.2 + 3.7 - 5.2} = 2^x$ $2^{10.9 - 5.2} = 2^x$ $2^{5.7} = 2^x$ * **Step 4:** Equate the powers since the bases on both sides are identical. $x = 5.7$ ### Exam Strategy & Shortcut To solve this rapidly, skip writing down the intermediate base steps. Mentally multiply the given powers by their base's relation to 2 (e.g., multiply 8's power by 3, and 16's power by 4). Just write down the final exponent string: $3(2.4) + 3.7 - 4(1.3) = 7.2 + 3.7 - 5.2$. This avoids rewriting the base $2$ repeatedly and directly yields $5.7$. ### Common Pitfall Students often mistake the division sign for a subtraction sign or misapply the division rule by dividing the exponents instead of subtracting them. Always remember that $a^m \div a^n$ means subtracting the exponent $n$ from $m$. ### Final Answer **Therefore, the correct answer is 5.7.**