Evaluate:
` sqrt (1.0816)`

Answer

To evaluate \( \sqrt{1.0816} \), we can follow these steps:

### Step 1: Remove the Decimal
We can express \( 1.0816 \) as \( \frac{10816}{100} \). This allows us to simplify the square root:

\[ \sqrt{1.0816} = \sqrt{\frac{10816}{100}} = \frac{\sqrt{10816}}{\sqrt{100}} = \frac{\sqrt{10816}}{10} \]

**Hint:** To simplify the square root of a decimal, convert it into a fraction.

### Step 2: Find the Square Root of 10816
Next, we need to find \( \sqrt{10816} \). We can use the long division method to find the square root of \( 10816 \).

1. Pair the digits of \( 10816 \) from right to left: \( 10 \) and \( 816 \).
2. Find the largest number whose square is less than or equal to \( 10 \). That number is \( 3 \) (since \( 3^2 = 9 \)).
3. Subtract \( 9 \) from \( 10 \) to get \( 1 \), and bring down the next pair \( 16 \) to make \( 116 \).

**Hint:** When pairing digits, always start from the decimal point or the rightmost side.

### Step 3: Continue the Long Division
Now, we need to find the next digit:

1. Double the quotient \( 3 \) to get \( 6 \).
2. Now, we need to find a digit \( x \) such that \( (60 + x) \times x \) is less than or equal to \( 116 \).
3. Testing \( x = 1 \): \( 61 \times 1 = 61 \) (valid).
4. Testing \( x = 2 \): \( 62 \times 2 = 124 \) (invalid).
5. So, we take \( x = 1 \). Now, subtract \( 61 \) from \( 116 \) to get \( 55 \) and bring down the next pair \( 00 \) to make \( 5500 \).

**Hint:** Always test digits starting from the largest possible value down to zero.

### Step 4: Repeat the Process
Now, we repeat the process:

1. Double the current quotient \( 31 \) to get \( 62 \).
2. Find \( x \) such that \( (620 + x) \times x \) is less than or equal to \( 5500 \).
3. Testing \( x = 8 \): \( 628 \times 8 = 5024 \) (valid).
4. Testing \( x = 9 \): \( 629 \times 9 = 5661 \) (invalid).
5. So, we take \( x = 8 \). Subtract \( 5024 \) from \( 5500 \) to get \( 476 \) and bring down \( 00 \) to make \( 47600 \).

**Hint:** Keep track of your calculations carefully to avoid errors.

### Step 5: Final Steps
1. Double the current quotient \( 318 \) to get \( 636 \).
2. Find \( x \) such that \( (6360 + x) \times x \) is less than or equal to \( 47600 \).
3. Testing \( x = 7 \): \( 6367 \times 7 = 44569 \) (valid).
4. Testing \( x = 8 \): \( 6368 \times 8 = 50944 \) (invalid).
5. So, we take \( x = 7 \). Subtract \( 44569 \) from \( 47600 \) to get \( 3021 \).

After completing the long division, we find that \( \sqrt{10816} = 104 \).

### Step 6: Final Calculation
Now, substitute back into our earlier expression:

\[ \sqrt{1.0816} = \frac{\sqrt{10816}}{10} = \frac{104}{10} = 10.4 \]

### Final Answer
Thus, \( \sqrt{1.0816} = 10.4 \).

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