Solve ((3×8)^5)^6: Multiple Exponentiation Expression

To solve the problem, we need to simplify the expression $\left(\left(3\times8\right)^5\right)^6$.

We will utilize the "power of a power" rule in exponents, which states $(a^m)^n = a^{m \times n}$. This rule tells us to multiply the exponents when raising a power to another power.

• Step 1: Identify the expression to simplify: $\left(\left(3 \times 8\right)^5\right)^6$.
• Step 2: Apply the power of a power rule: This gives us $(3 \times 8)^{5 \times 6}$.
• Step 3: Multiply the exponents: $5 \times 6 = 30$.

Therefore, the expression simplifies to $(3 \times 8)^{30}$.

Upon comparing this result with the provided answer choices, the correct choice is:

$\left(3\times8\right)^{5\times6}$

This choice correctly applies the power of a power rule, thereby validating the solution as correct.

In conclusion, the simplified form of the expression is $(3 \times 8)^{30}$, and the correct choice is option 4.