Evaluate the following without directly multiplying :
`196 xx 193`

Answer

To evaluate \( 196 \times 193 \) without directly multiplying, we can use the method of factorization. Here are the steps:

### Step-by-Step Solution:

1. **Rewrite the numbers**:
We can express \( 196 \) and \( 193 \) in terms of \( 200 \):
\[ 196 = 200 - 4 \]
\[ 193 = 200 - 7 \]

2. **Set up the expression**:
Now, we can rewrite the product:
\[ 196 \times 193 = (200 - 4)(200 - 7) \]

3. **Use the formula for the product of two binomials**:
We can use the formula \( (x - a)(x - b) = x^2 - (a + b)x + ab \). Here, \( x = 200 \), \( a = 4 \), and \( b = 7 \):
\[ = 200^2 - (4 + 7) \times 200 + (4 \times 7) \]

4. **Calculate each term**:
- Calculate \( 200^2 \):
\[ 200^2 = 40000 \]
- Calculate \( (4 + 7) \):
\[ 4 + 7 = 11 \]
- Calculate \( 4 \times 7 \):
\[ 4 \times 7 = 28 \]

5. **Substitute back into the expression**:
Now substitute these values back into the expression:
\[ = 40000 - 11 \times 200 + 28 \]

6. **Calculate \( 11 \times 200 \)**:
\[ 11 \times 200 = 2200 \]

7. **Combine all terms**:
Now, substitute this value back:
\[ = 40000 - 2200 + 28 \]
\[ = 40000 - 2200 = 37800 \]
\[ = 37800 + 28 = 37828 \]

### Final Answer:
Thus, the value of \( 196 \times 193 \) is \( 37828 \).

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