$8^7 \times 2^6 \div 8^{2.4} = 8^x$

### Concept & Formula This problem tests your ability to manipulate indices using both multiplication (addition of exponents) and division (subtraction of exponents) rules. $$ a^m \times a^n \div a^p = a^{m+n-p} $$ ### Step-by-Step Solution * Observe the target base on the right side of the equation, which is $8$. * Most terms are already in base $8$. We only need to convert $2^6$ into base $8$ to make calculation seamless. * Recall that $8 = 2^3$. Therefore, $2^6 = (2^3)^2 = 8^2$. * Substitute this back into the original equation: * $8^7 \times 8^2 \div 8^{2.4} = 8^x$ * Apply the laws of indices. Add the exponents for multiplication and subtract the exponent for division: * $8^{7 + 2 - 2.4} = 8^x$ * Perform the arithmetic on the exponents: * $7 + 2 = 9$ * $9 - 2.4 = 6.6$ * So, $8^{6.6} = 8^x$. Equating the exponents gives $x = 6.6$. ### Exam Strategy & Shortcut Keep your focus strictly on the target base ($8$). Do not waste time converting $8^7$ and $8^{2.4}$ into base $2$, as that creates unnecessarily large numbers and decimals. Just swap $2^6$ for $8^2$. The math then becomes a trivial mental calculation: $7 + 2 - 2.4 = 6.6$. ### Common Pitfall Converting all terms to base $2$ is a common trap. While mathematically valid (yielding $2^{21} \times 2^6 \div 2^{7.2} = 2^{3x}$), it takes significantly longer and involves solving $27 - 7.2 = 3x$, which introduces more opportunities for arithmetic errors under time pressure. ### Final Answer **Therefore, the correct answer is 6.6.**