Let's simplify the given algebraic expression step by step.
Given expression:
[tex]\[
\left(\frac{2 x^{10} y^5}{x^7 \cdot x^{-4}}\right)^3
\][/tex]
First, simplify the denominator inside the fraction:
[tex]\[
x^7 \cdot x^{-4} = x^{7 + (-4)} = x^{3}
\][/tex]
Now rewrite the expression with the simplified denominator:
[tex]\[
\left(\frac{2 x^{10} y^5}{x^3}\right)^3
\][/tex]
Next, simplify the fraction inside the brackets. Since the bases are the same (both are [tex]\(x\)[/tex]), subtract the exponents:
[tex]\[
\frac{2 x^{10} y^5}{x^3} = 2 x^{10 - 3} y^5 = 2 x^7 y^5
\][/tex]
Now, our expression looks like this:
[tex]\[
(2 x^7 y^5)^3
\][/tex]
Distribute the exponent 3 to each term inside the parentheses:
[tex]\[
(2^3) \cdot (x^7)^3 \cdot (y^5)^3
\][/tex]
Calculate each part separately:
[tex]\[
2^3 = 8
\][/tex]
[tex]\[
(x^7)^3 = x^{7 \cdot 3} = x^{21}
\][/tex]
[tex]\[
(y^5)^3 = y^{5 \cdot 3} = y^{15}
\][/tex]
Putting it all together, we get:
[tex]\[
8 x^{21} y^{15}
\][/tex]
Among the given options, our simplified expression [tex]\(8 x^{21} y^{15}\)[/tex] matches option (C).
Therefore, the correct answer is:
[tex]\[
\boxed{8 x^{21} y^{15}}
\][/tex]
