We are given the expression [tex]\(\sqrt[3]{200 k^{15}}\)[/tex] and we need to find an equivalent expression.
First, let's break down the components inside the cube root:
1. The number [tex]\(200\)[/tex] can be factored into its prime factors:
[tex]\[
200 = 2^3 \times 25
\][/tex]
2. The term [tex]\(k^{15}\)[/tex] can be expressed as:
[tex]\[
k^{15} = (k^5)^3
\][/tex]
Combining these, we can rewrite the original expression [tex]\(\sqrt[3]{200 k^{15}}\)[/tex] as:
[tex]\[
\sqrt[3]{2^3 \times 25 \times (k^5)^3}
\][/tex]
Now, using the property of cube roots that [tex]\(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\)[/tex], we can split the cube root into separate parts:
[tex]\[
\sqrt[3]{2^3 \times 25 \times (k^5)^3} = \sqrt[3]{2^3} \times \sqrt[3]{25} \times \sqrt[3]{(k^5)^3}
\][/tex]
We know that:
[tex]\[
\sqrt[3]{2^3} = 2 \quad \text{and} \quad \sqrt[3]{(k^5)^3} = k^5
\][/tex]
Thus, our expression simplifies to:
[tex]\[
2 \times k^5 \times \sqrt[3]{25}
\][/tex]
Therefore, the equivalent expression is:
[tex]\[
2 k^5 \sqrt[3]{25}
\][/tex]
Looking at the given options, we see that the correct answer is:
[tex]\[
\boxed{2}
\][/tex]
This corresponds to option B:
[tex]\[
2 k^5 \sqrt[3]{25}
\][/tex]
