Complete the Expression: Solving (10x)^8x Power Notation

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

• Step 1: Identify and restate the given expression.
• Step 2: Decide on the exponent technique to use.
• Step 3: Rewrite the expression using the identified rule.

Now, let's work through each step:
Step 1: The problem gives us the expression $(10x)^{8x}$.
Step 2: We will apply the power of a power rule for exponents, which states that if you have $(a^m)^n$, it equals $a^{m \cdot n}$.
Step 3: We need to express $8x$ as a product of two numbers. Let's write $8x$ as $(2x) \cdot 4$. Hence, using the power of a power rule, we can transform this into $((10x)^{2x})^4$.

Therefore, the solution to the problem is $(10x)^{8x} = ((10x)^{2x})^4$.

Upon examining the given answer choices, we find that choice 1: $\left(\left(10x\right)^{2x}\right)^4$ matches our solution. The other choices do not represent the original expression correctly using the power of a power rule.