To simplify the expression [tex]\( p^9 + q^9 + 3p^3q^3r^3 \)[/tex] given that [tex]\( p^3 + q^3 = r^3 \)[/tex], follow these steps:
1. Understand the given relation:
We know that:
[tex]\[
p^3 + q^3 = r^3
\][/tex]
2. Express powers [tex]\( p^9 \)[/tex] and [tex]\( q^9 \)[/tex]:
Write [tex]\( p^9 \)[/tex] and [tex]\( q^9 \)[/tex] in terms of [tex]\( p^3 \)[/tex] and [tex]\( q^3 \)[/tex]:
[tex]\[
p^9 = (p^3)^3
\][/tex]
[tex]\[
q^9 = (q^3)^3
\][/tex]
3. Rewrite the original expression:
Substitute these terms back into the original expression:
[tex]\[
p^9 + q^9 + 3 p^3 q^3 r^3 = (p^3)^3 + (q^3)^3 + 3 p^3 q^3 r^3
\][/tex]
4. Use the given relation in the expression:
Since [tex]\( r^3 = p^3 + q^3 \)[/tex], substitute [tex]\( r^3 \)[/tex] back into the expression:
[tex]\[
r^3 = p^3 + q^3
\][/tex]
5. Simplify the expression:
The expression simplifies by substituting [tex]\( r^3 \)[/tex] wherever [tex]\( p^3 + q^3 \)[/tex] appears:
[tex]\[
p^9 + q^9 + 3 p^3 q^3 r^3 = (p^3)^3 + (q^3)^3 + 3 p^3 q^3 (p^3 + q^3)
\][/tex]
6. Combine like terms:
Combine the terms involving [tex]\( p^3 \)[/tex] and [tex]\( q^3 \)[/tex]:
[tex]\[
(p^3)^3 + (q^3)^3 + 3 p^3 q^3 (p^3 + q^3) = p^3 p^6 + q^3 q^6 + 3 p^3 q^3 (p^3 + q^3)
\][/tex]
At this stage, the expression is simplified to:
[tex]\[
p^3 p^6 + q^3 q^6 + 3 p^3 q^3 (p^3 + q^3)
\][/tex]
Further, this can be read as:
[tex]\[
p^3 (p^3)^2 + q^3 (q^3)^2 + 3 p^3 q^3 (p^3 + q^3)
\][/tex]
So, the fully simplified form is:
[tex]\[
p^3 + 3p^3q^3(p^3 + q^3) + q^3
\][/tex]
Hence, the simplified expression is:
[tex]\[
p^3 + q^3 + 3p^3q^3(p^3 + q^3)
\][/tex]
This is the final simplified expression for [tex]\( p^9 + q^9 + 3p^3q^3r^3 \)[/tex].
