1. **State the problem:** Find the positive square root of 22.09. 2. **Recall the formula:** The square root of a number $x$ is a value $y$ such that $$y^2 = x$$. 3. **Apply the formula:** We want to find $y$ such that $$y^2 = 22.09$$. 4. **Calculate the square root:** Since 22.09 is a decimal, we can try to express it as a fraction or use a calculator method. 5. **Express 22.09 as a fraction:** $$22.09 = \frac{2209}{100}$$. 6. **Find the square root of numerator and denominator separately:** $$\sqrt{22.09} = \sqrt{\frac{2209}{100}} = \frac{\sqrt{2209}}{\sqrt{100}}$$. 7. **Calculate the square roots:** $$\sqrt{2209} = 47$$ (since $47 \times 47 = 2209$) and $$\sqrt{100} = 10$$. 8. **Simplify:** $$\frac{47}{10} = 4.7$$. 9. **Conclusion:** The positive square root of 22.09 is $$\boxed{4.7}$$.