Certainly! Let's break down the given mathematical expression:
[tex]\[
\left(x^{m-1} - 2.5\right) \left(x^{m-1} + 2.5\right)
\][/tex]
To simplify this, we can recognize that this expression is in the form of the difference of squares, which is a commonly used algebraic identity. The difference of squares states:
[tex]\[
(a - b)(a + b) = a^2 - b^2
\][/tex]
Here, we identify [tex]\( a \)[/tex] as [tex]\( x^{m-1} \)[/tex] and [tex]\( b \)[/tex] as 2.5. Applying the difference of squares identity, we get:
[tex]\[
\left(x^{m-1} - 2.5\right) \left(x^{m-1} + 2.5\right) = \left(x^{m-1}\right)^2 - (2.5)^2
\][/tex]
Now, let's calculate each term separately:
1. [tex]\(\left(x^{m-1}\right)^2 = x^{(m-1) \cdot 2} = x^{2(m-1)}\)[/tex]
2. [tex]\((2.5)^2 = 2.5 \cdot 2.5 = 6.25\)[/tex]
Substituting these back into our simplified expression, we have:
[tex]\[
x^{2(m-1)} - 6.25
\][/tex]
Therefore, the simplified form of [tex]\(\left(x^{m-1} - 2.5\right) \left(x^{m-1} + 2.5\right)\)[/tex] is:
[tex]\[
x^{2(m-1)} - 6.25
\][/tex]
