Evaluate:
`sqrt (10.0489)`

Answer

To evaluate \( \sqrt{10.0489} \), we can follow these steps:

### Step 1: Remove the Decimal
To make the calculation easier, we can remove the decimal by multiplying the number by 10,000 (which is \( 10^4 \)). This gives us:
\[ \sqrt{10.0489} = \sqrt{\frac{10.0489}{1}} = \sqrt{\frac{10.0489 \times 10000}{10000}} = \sqrt{100489} \div 100 \]

### Step 2: Pair the Digits
Next, we will pair the digits of 100489 from right to left:
- The pairs are: 1 | 00 | 48 | 9

### Step 3: Find the Largest Square
Now, we find the largest square less than or equal to the first pair (1):
- \( 1^2 = 1 \)
- Write 1 above the pair and subtract:
\[ 1 - 1 = 0 \]
Bring down the next pair (00), making it 00.

### Step 4: Double the Quotient
Double the quotient (which is currently 1):
- \( 2 \times 1 = 2 \)

### Step 5: Find the Next Digit
Now we need to find a digit \( x \) such that:
\[ (20 + x) \times x \leq 48 \]
Testing values:
- For \( x = 2 \): \( 22 \times 2 = 44 \) (this works)
- For \( x = 3 \): \( 23 \times 3 = 69 \) (this does not work)

So, we use \( x = 2 \):
- Write 2 above the pair and subtract:
\[ 48 - 44 = 4 \]
Bring down the next pair (89), making it 489.

### Step 6: Repeat the Process
Double the current quotient (which is now 12):
- \( 2 \times 12 = 24 \)

Now we need to find a digit \( y \) such that:
\[ (240 + y) \times y \leq 489 \]
Testing values:
- For \( y = 2 \): \( 242 \times 2 = 484 \) (this works)
- For \( y = 3 \): \( 243 \times 3 = 729 \) (this does not work)

So, we use \( y = 2 \):
- Write 2 above the pair and subtract:
\[ 489 - 484 = 5 \]

### Step 7: Combine the Results
Now we have:
- The quotient is 32 (from the digits we found: 1, 2, 2).
- Thus, \( \sqrt{100489} = 317 \).

### Step 8: Divide by 100
Finally, we divide by 100 to account for the decimal we removed:
\[ \sqrt{10.0489} = \frac{317}{100} = 3.17 \]

### Final Answer
\[ \sqrt{10.0489} = 3.17 \]

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