1. **Problem Statement:** We are given a table describing iterations of a fractal-like process with columns: Iteration number $n$, Number of Segments $4^n$, Length of Each Segment $81 \times (\frac{1}{3})^n$, and Total Length $4^n \times 81 \times (\frac{1}{3})^n$. 2. **Goal:** Understand and simplify the expression for the Total Length at iteration $n$. 3. **Formula and Rules:** The total length at iteration $n$ is given by: $$\text{Total Length} = 4^n \times 81 \times \left(\frac{1}{3}\right)^n$$ Recall the laws of exponents: - $a^m \times a^n = a^{m+n}$ - $(a^m)^n = a^{mn}$ 4. **Simplify the expression:** Rewrite the total length: $$4^n \times 81 \times \left(\frac{1}{3}\right)^n = 81 \times \left(4^n \times 3^{-n}\right) = 81 \times \left(\frac{4}{3}\right)^n$$ 5. **Interpretation:** The total length at iteration $n$ is: $$\boxed{81 \times \left(\frac{4}{3}\right)^n}$$ This shows the total length grows exponentially with base $\frac{4}{3}$ starting from 81 at iteration 0. 6. **Example check for $n=10$:** $$\text{Total Length} = 81 \times \left(\frac{4}{3}\right)^{10} = \frac{1048576}{729}$$ which matches the given value in the table. Thus, the total length formula is confirmed and simplified.