Square root of 11881 using long division method.

Steps to Solve

1. Group the digits from right to left Start by grouping the digits of 11881 into pairs from the right. For the number 11881, we have:

• 11 | 88 | 1

2. Find the largest square less than or equal to the first group Look at the first group, which is 11. The largest perfect square less than or equal to 11 is $3^2 = 9$.

3. Subtract and bring down the next group Subtract 9 from 11: $$ 11 - 9 = 2 $$ Now bring down the next group (88) next to 2, making it 288.

4. Double the quotient and find the next digit The current quotient is 3. Double this to get 6, and attempt to find a digit 'x' such that: $$ (60 + x) \cdot x \leq 288 $$ Testing with $x = 4$, we have: $$ (64) \cdot 4 = 256 $$ This works since $256 \leq 288$.

5. Subtract and bring down the last group Now subtract 256 from 288: $$ 288 - 256 = 32 $$ Next, bring down the last group (1), making it 321.

6. Double the quotient again and find the final digit The current quotient is 34. Double this to get 68, and find a digit 'y' such that: $$ (680 + y) \cdot y \leq 321 $$ Testing $y = 4$, we have: $$ (684) \cdot 4 = 2736 $$ This does not work. Testing $y = 3$, we have: $$ (683) \cdot 3 = 2049 $$ This also does not work. Testing $y = 2$, we get: $$ (682) \cdot 2 = 1364 $$ Again, it doesn't work. Testing $y = 1$, we have: $$ (681) \cdot 1 = 681 $$ Thus, we can’t proceed further with another digit.

7. Compile results The quotient so far is 109, so: $$ \sqrt{11881} \approx 109 $$