Evaluate (2×6×7×5×4×3×4)⁴: Complex Expression Challenge

To solve this problem, we'll follow these steps:

• Step 1: Identify the given expression
• Step 2: Apply the power of a product rule
• Step 3: Simplify and write the final expression

Now, let's work through each step:

Step 1: The given expression is $(2 \times 6 \times 7 \times 5 \times 4 \times 3 \times 4)$.
We are tasked with raising this entire product to the 4th power.

Step 2: The power of a product rule, $(a \times b)^n = a^n \times b^n$, allows us to distribute the power to each factor in the product.

Step 3: Distributing the exponent of 4 to each factor in the product:

• Raise 2 to the 4th power: $2^4$
• Raise 6 to the 4th power: $6^4$
• Raise 7 to the 4th power: $7^4$
• Raise 5 to the 4th power: $5^4$
• Raise 4 to the 4th power: $4^4$
• Raise 3 to the 4th power: $3^4$
• Raise the second 4 to the 4th power (as it appears twice in the original expression): $4^4$

By applying the exponent to each factor, the expression becomes:

$2^4 \times 6^4 \times 7^4 \times 5^4 \times 4^4 \times 3^4 \times 4^4$

Therefore, the expression $(2 \times 6 \times 7 \times 5 \times 4 \times 3 \times 4)^4$ is equal to: $2^4 \times 6^4 \times 7^4 \times 5^4 \times 4^4 \times 3^4 \times 4^4$.