Simplify : `(213 xx 213 + 187 xx 187)`

Options

A. A) 80338
B. B) 80438
C. C) 60338
D. D) 60438

Answer

To simplify the expression \( 213 \times 213 + 187 \times 187 \), we can use the formula for the sum of squares, which states that:

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

Here, let \( a = 213 \) and \( b = 187 \).

### Step 1: Calculate \( a + b \) and \( ab \)

First, we need to find \( a + b \) and \( ab \):

\[ a + b = 213 + 187 = 400 \]

\[ ab = 213 \times 187 \]

### Step 2: Calculate \( ab \)

To calculate \( ab \):

\[ 213 \times 187 = 213 \times (200 - 13) = 213 \times 200 - 213 \times 13 \]
\[ = 42600 - 2769 = 39831 \]

### Step 3: Substitute into the formula

Now substitute \( a + b \) and \( ab \) into the sum of squares formula:

\[ 213^2 + 187^2 = (400)^2 - 2 \times 39831 \]

### Step 4: Calculate \( (400)^2 \)

Calculate \( (400)^2 \):

\[ (400)^2 = 160000 \]

### Step 5: Calculate \( 2 \times 39831 \)

Now calculate \( 2 \times 39831 \):

\[ 2 \times 39831 = 79662 \]

### Step 6: Final Calculation

Now substitute back into the equation:

\[ 213^2 + 187^2 = 160000 - 79662 \]

Calculate the final result:

\[ 160000 - 79662 = 80338 \]

Thus, the simplified result of \( 213 \times 213 + 187 \times 187 \) is:

\[ \boxed{80338} \]