Evaluate `sqrt (16384).`

Options

A. 118
B. 138
C. 126
D. 128

Answer

To evaluate \( \sqrt{16384} \) using the long division method, we can follow these steps:

### Step 1: Pair the digits
Start from the right and pair the digits of the number. For \( 16384 \), we pair it as \( 16 | 38 | 4 \).

### Step 2: Find the largest square
Find the largest square number less than or equal to the first pair (16). The largest square is \( 4^2 = 16 \).

### Step 3: Subtract and bring down the next pair
Subtract \( 16 \) from \( 16 \) which gives \( 0 \). Bring down the next pair \( 38 \) to make it \( 038 \).

### Step 4: Double the quotient
Double the quotient (which is currently \( 4 \)) to get \( 8 \).

### Step 5: Find the next digit
Now we need to find a digit \( x \) such that \( (80 + x) \times x \) is less than or equal to \( 38 \).

- Trying \( x = 0 \): \( 80 \times 0 = 0 \) (valid)
- Trying \( x = 1 \): \( 81 \times 1 = 81 \) (too large)

So, we take \( x = 0 \), which gives us \( 80 \times 0 = 0 \).

### Step 6: Subtract and bring down the next pair
Now, we have \( 38 - 0 = 38 \). Bring down the next pair \( 04 \) to make it \( 3804 \).

### Step 7: Double the quotient again
Double the current quotient \( 40 \) (from \( 4 \)) to get \( 80 \).

### Step 8: Find the next digit again
Now we need to find a digit \( y \) such that \( (800 + y) \times y \) is less than or equal to \( 3804 \).

- Trying \( y = 4 \): \( 804 \times 4 = 3216 \) (valid)
- Trying \( y = 5 \): \( 805 \times 5 = 4025 \) (too large)

So, we take \( y = 4 \), which gives us \( 804 \times 4 = 3216 \).

### Step 9: Subtract and find the remainder
Now, \( 3804 - 3216 = 588 \).

### Step 10: Bring down the next pair
Since there are no more pairs to bring down, we can conclude our calculation.

### Final Step: Combine the results
The quotient we have calculated is \( 128 \), and since there is no remainder, we conclude that:

\[ \sqrt{16384} = 128 \]