To determine which of the given expressions is a sum of cubes, we need to examine each expression to see if it can be written in the form [tex]\( a^3 + b^3 \)[/tex].
Let's consider each expression in turn:
### Expression 1:
[tex]\[ -64 x^6 y^{12} + 125 x^{16} y^3 \][/tex]
We can rewrite this as:
[tex]\[ -(4x^2 y^4)^3 + (5x^5 y)^3 \][/tex]
This is not a sum of cubes but a difference of cubes.
### Expression 2:
[tex]\[ -32 x^6 y^{12} + 125 x^{16} y^3 \][/tex]
We can rewrite this as:
[tex]\[ -(2x^2 y^4)^3 + (5x^4 y)^3 \][/tex]
This is also a difference of cubes, not a sum of cubes.
### Expression 3:
[tex]\[ 32 x^6 y^{12} + 125 x^9 y^3 \][/tex]
We can rewrite this as:
[tex]\[ (2x^2 y^4)^3 + (5x^3 y)^3 \][/tex]
This is a sum of cubes of the form [tex]\( a^3 + b^3 \)[/tex], where [tex]\( a = 2x^2 y^4 \)[/tex] and [tex]\( b = 5x^3 y \)[/tex].
### Expression 4:
[tex]\[ 64 x^6 y^{12} + 125 x^9 y^3 \][/tex]
We can rewrite this as:
[tex]\[ (4x^2 y^4)^3 + (5x^3 y)^3 \][/tex]
This is also a sum of cubes of the form [tex]\( a^3 + b^3 \)[/tex], where [tex]\( a = 4x^2 y^4 \)[/tex] and [tex]\( b = 5x^3 y \)[/tex].
Upon verifying the forms carefully, the expressions:
[tex]\[
32 x^6 y^{12} + 125 x^9 y^3
\][/tex]
and
[tex]\[
64 x^6 y^{12} + 125 x^9 y^3
\][/tex]
can both be written as sums of cubes.
Since the problem only asks for one expression that fits the form of a sum of cubes,
[tex]\(\boxed{64 x^6 y^{12} + 125 x^9 y^3}\)[/tex]
the fourth choice, is the correct answer.
