To factor the expression [tex]\(125 a^6 - r^6 s^3\)[/tex] using the sum of cubes identity, we first need to express [tex]\(125 a^6\)[/tex] and [tex]\(r^6 s^3\)[/tex] in terms of cubes.
The identity for the difference of two cubes is:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
Let's identify the terms in the form of [tex]\(A^3\)[/tex] and [tex]\(B^3\)[/tex]:
1. Identify [tex]\(a^3\)[/tex] in the term [tex]\(125 a^6\)[/tex]:
[tex]\[
125 a^6 = (5a^2)^3
\][/tex]
Here, [tex]\(a = 5a^2\)[/tex].
2. Identify [tex]\(b^3\)[/tex] in the term [tex]\(r^6 s^3\)[/tex]:
[tex]\[
r^6 s^3 = (r^2 s)^3
\][/tex]
Here, [tex]\(b = r^2 s\)[/tex].
Now, substituting these back into the identity:
[tex]\[
a^3 - b^3 = (a - b)(a^2 + ab + b^2)
\][/tex]
In our case:
[tex]\[
(5a^2)^3 - (r^2 s)^3
\][/tex]
Thus, we have:
[tex]\[
A = 5a^2 \quad \text{and} \quad B = r^2 s
\][/tex]
So, the values are:
[tex]\[
a = 5a^2 \quad \text{and} \quad b = r^2 s
\][/tex]
