Question: User: $\left(\sqrt{223109}\right)$ /101 simplified
1. **State the problem:** Simplify the expression $$\frac{\sqrt{223109}}{101}$$.
2. **Recall the formula:** To simplify a fraction with a square root in the numerator, check if the number inside the square root is a perfect square or can be factored to include a perfect square.
3. **Check if 223109 is a perfect square:** Calculate $$101^2 = 10201$$, which is much smaller than 223109, so 223109 is not equal to $$101^2$$.
4. **Try to express 223109 as a product involving 101:**
Calculate $$223109 \div 101 = 2209$$.
5. **Check if 2209 is a perfect square:**
Calculate $$47^2 = 2209$$, so yes, 2209 is a perfect square.
6. **Rewrite the expression:**
$$\frac{\sqrt{223109}}{101} = \frac{\sqrt{101 \times 2209}}{101} = \frac{\sqrt{101} \times \sqrt{2209}}{101} = \frac{\sqrt{101} \times 47}{101}$$
7. **Simplify the fraction:**
$$\frac{\sqrt{101} \times 47}{101} = 47 \times \frac{\sqrt{101}}{101}$$
8. **Express the final simplified form:**
$$\boxed{\frac{47 \sqrt{101}}{101}}$$
This is the simplified form of the original expression.