If `7^(2) =49, 67^(2)=4489`, then `667^(2)=`________. Options are (a) 448844 (b) 444088 (c) 444888 (d) 444889 .

Options

A. 448844
B. 444088
C. 444888
D. 444889

Answer

To find the value of \( 667^2 \), we can use the algebraic identity for the square of a binomial, which states that:

\[ (a + b)^2 = a^2 + 2ab + b^2 \]

Here, we can break down \( 667 \) into \( 600 + 67 \).

### Step 1: Identify \( a \) and \( b \)
Let:
- \( a = 600 \)
- \( b = 67 \)

### Step 2: Calculate \( a^2 \)
Now, calculate \( a^2 \):
\[ 600^2 = (6 \times 10^2)^2 = 36 \times 10^4 = 360000 \]

### Step 3: Calculate \( b^2 \)
Next, calculate \( b^2 \):
\[ 67^2 = 4489 \quad \text{(given in the problem)} \]

### Step 4: Calculate \( 2ab \)
Now, calculate \( 2ab \):
\[ 2 \times 600 \times 67 = 2 \times 600 \times 67 = 80400 \]

### Step 5: Combine the results
Now, combine all the results using the formula:
\[ 667^2 = a^2 + 2ab + b^2 = 360000 + 80400 + 4489 \]

### Step 6: Perform the addition
Now, let's add these values together:
\[ 360000 + 80400 = 440400 \]
\[ 440400 + 4489 = 444889 \]

Thus, the value of \( 667^2 \) is:
\[ \boxed{444889} \]