To factor the given expression [tex]\(64v^2 - f^2\)[/tex] completely, we can use the difference of squares method. The difference of squares formula is given by:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
In this case, our expression [tex]\(64v^2 - f^2\)[/tex] fits the difference of squares pattern, where:
- [tex]\(a^2 = 64v^2\)[/tex]
- [tex]\(b^2 = f^2\)[/tex]
First, we identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] from the given expression:
- Since [tex]\(64v^2\)[/tex] is a perfect square, we can write it as [tex]\((8v)^2\)[/tex]. Therefore, [tex]\(a = 8v\)[/tex].
- Similarly, since [tex]\(f^2\)[/tex] is a perfect square, we have [tex]\(b = f\)[/tex].
Using the difference of squares formula:
[tex]\[
(8v)^2 - f^2 = (8v - f)(8v + f)
\][/tex]
Therefore, the completely factored form of the expression [tex]\(64v^2 - f^2\)[/tex] is:
[tex]\[
(8v - f)(8v + f)
\][/tex]
