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$$1 + \frac{x}{100} = 1.25^{\frac{1}{6}}.$$\n\n5. Express $x$ explicitly:\n$$\frac{x}{100} = 1.25^{\frac{1}{6}} - 1.$$\nMultiply both sides by 100:\n$$x = 100 \left(1.25^{\frac{1}{6}} - 1\right).$$\n\n6. Final answer: Evaluate numerically:\nCalculate $1.25^{\frac{1}{6}} \approx 1.03797$, so\n$$x \approx 100 \times (1.03797 - 1) = 100 \times 0.03797 = 3.797.$$\nTherefore, the rate $x$ is approximately $3.80$%.