The difference of two squares can be represented by the formula:
\[ a^2 - b^2 = (a - b)(a + b) \]
To express \(13 \times 19\) using the difference of two squares, we can rewrite it in a suitable form. Notice that:
\[ 13 \times 19 = 19^2 - 6^2 \]
Here’s how it works:
- Recognize that \(19\) can be seen as \( (16 + 3) \) and \(13\) can be seen as the distance from \(16\) to \(19\).
- This can be expressed as:
\[ 19^2 - 6^2 = (19 - 6)(19 + 6) \]
Calculating the squares:
- \(19^2 = 361\)
- \(6^2 = 36\)
Thus:
\[ 19^2 - 6^2 = 361 - 36 = 325 \]
Noticing that \( 13 \times 19 = 247\) suggests that a reconsideration in terms of squares is necessary.
To rewrite it correctly,
\[ (16)^2 - (3)^2 = 256 - 9 = 247 \]
We can also write:
\[ 13 = 16 - 3 \quad \text{and} \quad 19 = 16 + 3 \]
Thus:
\[ (16 - 3)(16 + 3) = 13 \times 19 \]
Therefore, \(13 \times 19 = (16 - 3)(16 + 3) = 16^2 - 3^2\).
So, \(13 \times 19\) can be rewritten using the difference of two squares as:
\[ 16^2 - 3^2 \]
This keeps \(16\) and \(3\) as roles in our targeting components for differentiation.
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