Find the value of the following expression:
`101xx101+99xx99`

Options

A. 40001
B. 30003
C. 20002
D. 20001

Answer

To solve the expression \( 101 \times 101 + 99 \times 99 \), we can use the algebraic identities for squares. Let's break it down step by step:

### Step 1: Rewrite the expression using algebraic identities
We can express \( 101 \) and \( 99 \) in terms of \( 100 \):
\[ 101 = 100 + 1 \quad \text{and} \quad 99 = 100 - 1 \]
Thus, we can rewrite the expression as:
\[ (100 + 1)^2 + (100 - 1)^2 \]

### Step 2: Apply the square of a binomial formula
Using the formula \( (a + b)^2 = a^2 + 2ab + b^2 \) and \( (a - b)^2 = a^2 - 2ab + b^2 \), we can expand both squares:
\[ (100 + 1)^2 = 100^2 + 2 \cdot 100 \cdot 1 + 1^2 = 10000 + 200 + 1 = 10201 \]
\[ (100 - 1)^2 = 100^2 - 2 \cdot 100 \cdot 1 + 1^2 = 10000 - 200 + 1 = 9801 \]

### Step 3: Add the results of the expansions
Now, we add the two results together:
\[ 10201 + 9801 \]

### Step 4: Perform the addition
Calculating the sum:
\[ 10201 + 9801 = 20002 \]

### Final Answer
Thus, the value of the expression \( 101 \times 101 + 99 \times 99 \) is:
\[ \boxed{20002} \]