Find the value of `sqrt(4.2436)`
(a)2.04
(b)2.14
(C)2.06
(d)2.16

Options

A. 2.04
B. 2.14
C. 2.06
D. 2.16

Answer

To find the value of \( \sqrt{4.2436} \), we can follow these steps:

### Step 1: Remove the Decimal Point
First, we can eliminate the decimal point by expressing \( 4.2436 \) in a form that allows us to work with whole numbers. Since there are four digits after the decimal point, we can multiply the number by \( 10^4 \) (which is 10,000) to convert it into a whole number.

\[ 4.2436 \times 10^4 = 42436 \]

### Step 2: Set Up for Square Root Calculation
Now we need to find \( \sqrt{42436} \). To do this, we can use the long division method for square roots.

### Step 3: Pair the Digits
Starting from the decimal point, we pair the digits of \( 42436 \) from right to left. This gives us the pairs: \( 36 \) and \( 424 \).

### Step 4: Find the Largest Square
We start with the leftmost pair, \( 424 \). The largest square less than or equal to \( 424 \) is \( 20^2 = 400 \). So, we write down \( 20 \) as the first part of our answer.

\[ 20 \quad (20^2 = 400) \]

### Step 5: Subtract and Bring Down the Next Pair
Now we subtract \( 400 \) from \( 424 \):

\[ 424 - 400 = 24 \]

Next, we bring down the next pair, which is \( 36 \), making it \( 2436 \).

### Step 6: Double the Current Quotient
Next, we double the current quotient \( 20 \) to get \( 40 \). We need to find a digit \( x \) such that \( (40 + x) \times x \) is less than or equal to \( 2436 \).

### Step 7: Trial and Error for the Next Digit
We can try different values for \( x \):

- For \( x = 6 \):
\[ (40 + 6) \times 6 = 46 \times 6 = 276 \quad (\text{too low}) \]
- For \( x = 7 \):
\[ (40 + 7) \times 7 = 47 \times 7 = 329 \quad (\text{too low}) \]
- For \( x = 8 \):
\[ (40 + 8) \times 8 = 48 \times 8 = 384 \quad (\text{too low}) \]
- For \( x = 9 \):
\[ (40 + 9) \times 9 = 49 \times 9 = 441 \quad (\text{too low}) \]
- For \( x = 10 \):
\[ (40 + 10) \times 10 = 50 \times 10 = 500 \quad (\text{too high}) \]

The largest \( x \) that works is \( 6 \), so we write \( 6 \) as the next digit of our quotient.

### Step 8: Calculate the Remainder
Now we calculate:

\[ 2436 - 276 = 2160 \]

### Step 9: Finalize the Result
Since we have \( 206 \) as our final quotient from the long division, we can express this as:

\[ \sqrt{42436} = 206 \]

### Step 10: Adjust for the Decimal Point
Since we multiplied by \( 10^4 \) initially, we need to divide our result by \( 100 \) (or \( 10^2 \)) to adjust for the decimal point:

\[ \sqrt{4.2436} = \frac{206}{100} = 2.06 \]

### Final Answer
Thus, the value of \( \sqrt{4.2436} \) is \( 2.06 \), which corresponds to option (c).

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