Find the square root of :
`5.2005` correct to two places of decimal.

Answer

To find the square root of 5.2005 correct to two decimal places, 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 of the number from the decimal point. For 5.2005, we can pair it as follows:
- Whole number part: 5
- Decimal part: 20 | 05

### Step 2: Find the largest square
Now, we need to find the largest square number less than or equal to the first pair (5). The largest square less than or equal to 5 is 2² = 4.

### Step 3: Subtract and bring down the next pair
Subtract 4 from 5:
\[ 5 - 4 = 1 \]
Now, bring down the next pair (20), making it 120.

### Step 4: Double the quotient
Double the quotient we found (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 (let's call it x) such that:
\[ (40 + x) \times x \leq 120 \]
Testing values:
- For x = 2: \( 42 \times 2 = 84 \) (this works)
- For x = 3: \( 43 \times 3 = 129 \) (this is too high)

So, we take x = 2. Now, our quotient is 2.2.

### Step 6: Subtract and bring down the next pair
Subtract 84 from 120:
\[ 120 - 84 = 36 \]
Now, bring down the next pair (05), making it 3605.

### Step 7: Double the current quotient
Double the current quotient (which is 22) to get 44.

### Step 8: Find the next digit
Now, we need to find a digit (let's call it y) such that:
\[ (440 + y) \times y \leq 3605 \]
Testing values:
- For y = 8: \( 448 \times 8 = 3584 \) (this works)
- For y = 9: \( 449 \times 9 = 4041 \) (this is too high)

So, we take y = 8. Now our quotient is 2.28.

### Step 9: Subtract and finalize
Subtract 3584 from 3605:
\[ 3605 - 3584 = 21 \]
Since we are asked for the square root correct to two decimal places, we can round our answer.

Thus, the square root of 5.2005 correct to two decimal places is:
\[ \sqrt{5.2005} \approx 2.28 \]