To determine the expression equivalent to [tex]\(64 y^{18}-1000 z^6\)[/tex], we must rewrite the given expression in the form of [tex]\(a^3 - b^3\)[/tex]. Let's break this down step-by-step:
1. Identify possible values for [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
Start by recognizing that the given expression is in the form [tex]\(64y^{18} - 1000z^6\)[/tex].
We want to express both terms as cubes:
[tex]\[
a^3 = 64y^{18} \quad \text{and} \quad b^3 = 1000z^6
\][/tex]
2. Express [tex]\(64y^{18}\)[/tex] as a cube:
Notice that:
[tex]\[
64 = 4^3 \quad \text{and} \quad y^{18} = (y^6)^3
\][/tex]
Hence, we can combine them as:
[tex]\[
64y^{18} = (4y^6)^3
\][/tex]
3. Express [tex]\(1000z^6\)[/tex] as a cube:
Notice that:
[tex]\[
1000 = 10^3 \quad \text{and} \quad z^6 = (z^2)^3
\][/tex]
Hence, we can combine them as:
[tex]\[
1000z^6 = (10z^2)^3
\][/tex]
4. Combining both expressions:
Now we have:
[tex]\[
64y^{18} = (4y^6)^3 \quad \text{and} \quad 1000z^6 = (10z^2)^3
\][/tex]
Therefore:
[tex]\[
64y^{18} - 1000z^6 = (4y^6)^3 - (10z^2)^3
\][/tex]
Thus, the expression equivalent to [tex]\(64 y^{18} - 1000 z^6\)[/tex] is:
[tex]\[
\left(4 y^6\right)^3 - \left(10 z^2\right)^3
\][/tex]
Hence, the correct choice is:
[tex]\[
\boxed{\left(4 y^6\right)^3 - \left(10 z^2\right)^3}
\][/tex]
