Simplify (8×2)³/(2×8)⁷: Exponential Division Challenge

To solve this problem, follow these steps:

• Step 1: Simplify the expression using exponent rules.
• Step 2: Compare with the given answer choices.

Step 1: Simplify the expression $\frac{(8 \times 2)^3}{(2 \times 8)^7}$.
Note that the bases in both the numerator and denominator are identical: $8 \times 2 = 16$. So the expression can be rewritten as:

$\frac{16^3}{16^7}$.

Using the exponent division rule, $\frac{a^m}{a^n} = a^{m-n}$, simplify the expression:

$16^{3-7} = 16^{-4}$.

According to the negative exponent rule, $a^{-n} = \frac{1}{a^n}$, this becomes:

$\frac{1}{16^4}$.

Step 2: Compare the simplification $\frac{1}{16^4}$ to the answer choices:

• Choice 1: $(8 \times 2)^{-4}$: Indicates $16^{-4}$, which is correct.
• Choice 2: $\frac{1}{(8 \times 2)^4}$: Equivalent to $\frac{1}{16^4}$, which is also correct.
• Choice 3: $\frac{1}{(8 \times 2)^{-4}}$: Implies $16^4$, which is incorrect.
• Choice 4: $a'+b'$ are correct: Is correct since both A and B represent the same solution.

Therefore, the correct choice is Choice 4: a'+b' are correct.

I'm confident in this solution as it accurately applies the rules of exponents. All recalculations confirm the analysis and answers provided.