Calculate (8×9×10×5×6×3×7)⁵: Complex Exponent Problem

To solve this problem, we need to appropriately apply the power of a product rule:

• Preliminary setup: Identify the expression within the power. We have $8\times9\times10\times5\times6\times3\times7$ raised to the 5th power.
• Step 1: Apply the exponent to each individual factor according to the power of a product rule:
  $\left(8\times9\times10\times5\times6\times3\times7\right)^5 = 8^5 \times 9^5 \times 10^5 \times 5^5 \times 6^5 \times 3^5 \times 7^5$
  • Here, each factor inside the parentheses is raised to the 5th power individually.
• Step 2: Compare this expanded expression to the choices provided.
• Choice 1: $8^5 \times 9^5 \times \left(10\times5\times6\times3\times7\right)$ does not raise each to the 5th power correctly.
• Choice 2: $8 \times 9 \times 10^5 \times 5^5 \times 6^5 \times 3^5 \times 7^5$ incorrectly multiplies 8 and 9 without raising them to the 5th power, so it is not correct.
• Choice 3: $5 \times \left(8\times9\times10\times5\times6\times3\times7\right)$ does not involve the correct exponentiation.
• Choice 4: "None of the answers are correct" seems accurate since none of the expressions match what we derived.

Therefore, after verifying all the choices, the correct answer is None of the answers are correct, based on the power of a product rule.