1. **State the problem:** Juan invested 23000 six years ago, and now the investment is worth 42557. We need to find the effective annual rate of return $r$ over 6 years. 2. **Formula used:** The value of an investment with compound interest is given by: $$A = P(1 + r)^t$$ where $A$ is the amount after $t$ years, $P$ is the principal, $r$ is the annual rate of return, and $t$ is the time in years. 3. **Plug in known values:** $$42557 = 23000(1 + r)^6$$ 4. **Isolate $(1 + r)^6$:** $$\frac{42557}{23000} = (1 + r)^6$$ 5. **Simplify the fraction:** $$\frac{\cancel{42557}}{\cancel{23000}} = (1 + r)^6$$ Numerically, $$1.8503 = (1 + r)^6$$ 6. **Take the 6th root of both sides to solve for $1 + r$:** $$1 + r = \sqrt[6]{1.8503}$$ 7. **Calculate the 6th root:** $$1 + r \approx 1.1073$$ 8. **Solve for $r$:** $$r = 1.1073 - 1 = 0.1073$$ 9. **Convert to percentage and round:** $$r = 10.73\%$$ **Final answer:** The effective annual rate of return is approximately **10.73%**.