Find the value of the following expression.
`106xx106+94xx94`

Options

A. 20036
B. 20072
C. 20049
D. 20098

Answer

To find the value of the expression \( 106 \times 106 + 94 \times 94 \), we can follow these steps:

### Step 1: Rewrite the expression using squares
We can express the products as squares:
\[ 106 \times 106 = 106^2 \]
\[ 94 \times 94 = 94^2 \]
So, the expression becomes:
\[ 106^2 + 94^2 \]

### Step 2: Use the identity for squares
We can use the identity \( a^2 + b^2 = (a + b)^2 - 2ab \) to simplify the calculation. Here, let \( a = 106 \) and \( b = 94 \).

### Step 3: Calculate \( a + b \) and \( ab \)
First, calculate \( a + b \):
\[ a + b = 106 + 94 = 200 \]
Next, calculate \( ab \):
\[ ab = 106 \times 94 \]

### Step 4: Calculate \( ab \) using the distributive property
To calculate \( 106 \times 94 \), we can break it down:
\[ 106 \times 94 = 106 \times (100 - 6) = 10600 - 636 = 9994 \]

### Step 5: Substitute into the identity
Now substitute \( a + b \) and \( ab \) into the identity:
\[ 106^2 + 94^2 = (200)^2 - 2 \times 9994 \]

### Step 6: Calculate \( (200)^2 \)
Calculating \( (200)^2 \):
\[ (200)^2 = 40000 \]

### Step 7: Calculate \( 2 \times 9994 \)
Now calculate \( 2 \times 9994 \):
\[ 2 \times 9994 = 19988 \]

### Step 8: Final calculation
Now, substitute back into the equation:
\[ 106^2 + 94^2 = 40000 - 19988 = 20012 \]

### Step 9: Verify the calculation
We can also calculate \( 106^2 \) and \( 94^2 \) directly:
\[ 106^2 = 11236 \]
\[ 94^2 = 8836 \]
Adding these:
\[ 11236 + 8836 = 20072 \]

Thus, the final answer is:
\[ \boxed{20072} \]