Question

how can 13x19 be rewritten using the difference of two squares

Answer 1

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:

1. Recognize that \(19\) can be seen as \( (16 + 3) \) and \(13\) can be seen as the distance from \(16\) to \(19\).
2. This can be expressed as:

\[
19^2 - 6^2 = (19 - 6)(19 + 6)
\]

Calculating the squares:

1. \(19^2 = 361\)
2. \(6^2 = 36\)

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.