$(64)^4 \div (8)^5 = x$

## Concept & Strategy To divide exponential terms with different bases, you must express the larger base in terms of the smaller base if possible. Once the bases are matched, apply the quotient rule of indices. $$ (a^m)^n = a^{m \times n} $$ $$ \frac{a^m}{a^n} = a^{m - n} $$ ## Step-by-Step Solution * **Given:** The expression is $(64)^4 \div (8)^5$. * **Calculation / Deduction:** * We can observe that the larger base ($64$) is a perfect square of the smaller base ($8$). * Express $64$ as a power of $8$: $$ 64 = 8^2 $$ * Substitute this back into the original expression: $$ (8^2)^4 \div 8^5 $$ * Apply the power of a power rule to the first term: $$ 8^{2 \times 4} \div 8^5 $$ $$ 8^8 \div 8^5 $$ * Now, apply the quotient rule of indices: $$ 8^{8 - 5} = 8^3 $$ * Look at the given options: (a) $(8)^8$, (b) $(8)^2$, (c) $(8)^{12}$, (d) $(8)^4$. * The calculated answer $8^3$ is not present in the options. ## Exam Strategy & Shortcut Use mental math. Instantly recognize $64$ as $8^2$. Mentally multiply the exponent $2 \times 4$ to get $8$. Then subtract the dividing exponent $5$. $8 - 5 = 3$. The answer is $8^3$. Since it's not an option, pick (e) immediately without putting pen to paper. ## Common Pitfall A classic mistake is dividing the bases directly without properly resolving the exponents, falsely assuming $64 \div 8 = 8$ and then guessing the exponent. Another error is doubting your own correct calculation of $8^3$ simply because it isn't among the first four options. Trust your math. ## Final Answer **Therefore, the correct answer is None of these.**