Find the square root of :
`531.7636`

Answer

To find the square root of `531.7636`, we can use the long division method for square roots. Here’s a step-by-step solution:

### Step 1: Pair the digits
Start by pairing the digits from the decimal point. For the number `531.7636`, we can pair it as follows:
- Before the decimal: `5 | 31`
- After the decimal: `76 | 36`

So, we have pairs: `5 | 31 | 76 | 36`.

### Step 2: Find the largest square
Find the largest square less than or equal to the first pair (5). The largest square is `2^2 = 4`.

### Step 3: Subtract and bring down
- Write `2` above the line (this is the first digit of the square root).
- Subtract `4` from `5`, which gives us `1`.
- Bring down the next pair (31), making it `131`.

### Step 4: Double the root
Double the number above the line (which is `2`), giving us `4`. We will use this to find the next digit of the square root.

### Step 5: Find the next digit
Now we need to find a digit `x` such that:
\[ 4x \times x \leq 131 \]

Testing values:
- For `x = 3`: \( 43 \times 3 = 129 \) (this works)
- For `x = 4`: \( 44 \times 4 = 176 \) (this is too high)

So, we take `3` as the next digit.

### Step 6: Subtract and bring down
- Write `3` above the line next to `2`, making it `23`.
- Subtract `129` from `131`, which gives us `2`.
- Bring down the next pair (76), making it `276`.

### Step 7: Double the root again
Double the current root (which is `23`), giving us `46`.

### Step 8: Find the next digit
Now we need to find a digit `y` such that:
\[ 46y \times y \leq 276 \]

Testing values:
- For `y = 5`: \( 465 \times 5 = 2325 \) (too high)
- For `y = 4`: \( 464 \times 4 = 1856 \) (too high)
- For `y = 3`: \( 463 \times 3 = 1389 \) (too high)
- For `y = 2`: \( 462 \times 2 = 924 \) (too high)
- For `y = 1`: \( 461 \times 1 = 461 \) (too high)
- For `y = 0`: \( 460 \times 0 = 0 \) (this works)

So we take `0` as the next digit.

### Step 9: Subtract and bring down
- Write `0` above the line next to `23`, making it `230`.
- Subtract `0` from `276`, which gives us `276`.
- Bring down the next pair (36), making it `27636`.

### Step 10: Double the root again
Double the current root (which is `230`), giving us `460`.

### Step 11: Find the next digit
Now we need to find a digit `z` such that:
\[ 460z \times z \leq 27636 \]

Testing values:
- For `z = 6`: \( 4606 \times 6 = 27636 \) (this works)

So, we take `6` as the next digit.

### Step 12: Final subtraction
- Write `6` above the line next to `230`, making it `2306`.
- Subtract `27636` from `27636`, which gives us `0`.

### Conclusion
The square root of `531.7636` is `23.06`.

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