Simplify: `(213xx213+187xx187)`

Options

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

Answer

To simplify the expression \( 213 \times 213 + 187 \times 187 \), we can follow these steps:

### Step 1: Rewrite the expression using squares
We can express the terms as squares:
\[ 213 \times 213 = 213^2 \]
\[ 187 \times 187 = 187^2 \]
So, the expression becomes:
\[ 213^2 + 187^2 \]

### Step 2: Use the identity for the sum of squares
We can use the identity:
\[ A^2 + B^2 = (A + B)^2 - 2AB \]
Here, let \( A = 213 \) and \( B = 187 \). Therefore, we can rewrite the expression as:
\[ 213^2 + 187^2 = (213 + 187)^2 - 2 \times 213 \times 187 \]

### Step 3: Calculate \( A + B \)
Now, calculate \( A + B \):
\[ 213 + 187 = 400 \]

### Step 4: Calculate \( (A + B)^2 \)
Now, calculate \( (A + B)^2 \):
\[ 400^2 = 160000 \]

### Step 5: Calculate \( 2AB \)
Next, calculate \( 2 \times 213 \times 187 \):
\[ 213 \times 187 = 39831 \]
So,
\[ 2 \times 39831 = 79662 \]

### Step 6: Substitute back into the identity
Now substitute back into the identity:
\[ 213^2 + 187^2 = 160000 - 79662 \]

### Step 7: Perform the subtraction
Now, perform the subtraction:
\[ 160000 - 79662 = 80338 \]

### Final Answer
Thus, the simplified expression \( 213 \times 213 + 187 \times 187 \) equals:
\[ \boxed{80338} \]