Evaluate `103xx97` by using identities .

Answer

To evaluate \( 103 \times 97 \) using identities, we can apply the difference of squares formula. Here’s a step-by-step solution:

### Step 1: Rewrite the expression
We can express \( 103 \) and \( 97 \) in terms of \( 100 \):
\[ 103 = 100 + 3 \quad \text{and} \quad 97 = 100 - 3 \]
Thus, we can rewrite the product as:
\[ 103 \times 97 = (100 + 3)(100 - 3) \]

### Step 2: Identify the identity
Recognizing this as a form of the difference of squares, we can use the identity:
\[ A^2 - B^2 = (A + B)(A - B) \]
Here, \( A = 100 \) and \( B = 3 \).

### Step 3: Apply the identity
Using the identity, we can simplify:
\[ (100 + 3)(100 - 3) = 100^2 - 3^2 \]

### Step 4: Calculate \( 100^2 \) and \( 3^2 \)
Now, we calculate:
\[ 100^2 = 10000 \quad \text{and} \quad 3^2 = 9 \]

### Step 5: Subtract the squares
Now we substitute back into the equation:
\[ 100^2 - 3^2 = 10000 - 9 = 9991 \]

### Final Answer
Thus, the value of \( 103 \times 97 \) is:
\[ \boxed{9991} \]

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