1. **State the problem:** Solve for $x$ in the equation $$100000 = 80000 \left(1 + \frac{x}{100}\right)^6.$$\n\n2. **Formula and explanation:** This is a compound growth equation where the amount after 6 periods is given by $$A = P \left(1 + \frac{r}{100}\right)^n,$$ where $A$ is the final amount, $P$ is the principal, $r$ is the rate in percent, and $n$ is the number of periods. We need to find $x$ which represents the rate $r$.\n\n3. **Isolate the growth factor:** Divide both sides by 80000:\n$$\frac{100000}{80000} = \left(1 + \frac{x}{100}\right)^6.$$\nSimplify the left side:\n$$1.25 = \left(1 + \frac{x}{100}\right)^6.$$\n\n4. **Take the sixth root:** To solve for $1 + \frac{x}{100}$, take the sixth root of both sides:\n$$\left(1 + \frac{x}{100}\right) = 1.25^{\frac{1}{6}}.$$\n\n5. **Express $x$ explicitly:**\n$$1 + \frac{x}{100} = 1.25^{\frac{1}{6}} \implies \frac{x}{100} = 1.25^{\frac{1}{6}} - 1.$$\nMultiply both sides by 100:\n$$x = 100 \left(1.25^{\frac{1}{6}} - 1\right).$$\n\n6. **Final answer:** The exact value of $x$ is $$x = 100 \left(1.25^{\frac{1}{6}} - 1\right).$$\nThis expression can be evaluated numerically if needed, but as requested, we leave it unrounded.