Sqrt 169 57Ab08

1. **State the problem:** We need to find the square root of $169$ using the long division method, as indicated by the notation $9|169$. 2. **Formula and rules:** The long division method for square roots involves pairing digits from right to left, finding the largest digit whose square is less than or equal to the first pair, subtracting, bringing down the next pair, doubling the current quotient, and finding the next digit. 3. **Step-by-step solution:** - Pair the digits of $169$ as $1|69$. - Find the largest number whose square is less than or equal to $1$. That is $1$ because $1^2=1$. - Write $1$ as the first digit of the root. - Subtract $1^2=1$ from $1$, remainder $0$. - Bring down the next pair $69$, making the new dividend $069$. - Double the current root $1$ to get $2$. - Find a digit $x$ such that $2x \times x \leq 69$. - Try $x=3$: $23 \times 3 = 69$, which fits exactly. - Write $3$ as the next digit of the root. - Subtract $69$ from $69$, remainder $0$. - No more pairs to bring down, remainder is $0$, so the root is complete. 4. **Final answer:** $$\sqrt{169} = 13$$ 5. **Check the answer:** Calculate $13^2 = 169$, which matches the original number. Therefore, the solution is correct.