To solve the problem of determining which of the given choices is equal to the expression [tex]\((27 \cdot 250)^{\frac{1}{3}}\)[/tex], we analyze it step by step:
1. Evaluate the Original Expression:
First, we need to simplify the expression [tex]\((27 \cdot 250)^{\frac{1}{3}}\)[/tex].
2. Break Down the Expression:
[tex]\[
27 \cdot 250
\][/tex]
Instead of calculating the product [tex]\(27 \cdot 250\)[/tex] directly, let's recognize that:
[tex]\[
27 = 3^3 \quad \text{and} \quad 250 = 2 \cdot 125 = 2 \cdot 5^3
\][/tex]
Therefore,
[tex]\[
27 \cdot 250 = 3^3 \cdot 2 \cdot 5^3
\][/tex]
3. Combine and Simplify Under the Cube Root:
Given the product inside the cube root, we have:
[tex]\[
(3^3 \cdot 2 \cdot 5^3)^{\frac{1}{3}}
\][/tex]
Using the properties of exponents, we can split the terms within the cube root:
[tex]\[
(3^3)^{\frac{1}{3}} \cdot (2)^{\frac{1}{3}} \cdot (5^3)^{\frac{1}{3}}
\][/tex]
4. Simplify Each Term:
Since [tex]\((a^b)^{\frac{1}{b}} = a\)[/tex], we know:
[tex]\[
(3^3)^{\frac{1}{3}} = 3
\][/tex]
Similarly,
[tex]\[
(5^3)^{\frac{1}{3}} = 5
\][/tex]
So we are left with:
[tex]\[
3 \cdot 5 \cdot (2)^{\frac{1}{3}}
\][/tex]
Simplify this further:
[tex]\[
3 \cdot 5 = 15
\][/tex]
Thus, the expression is:
[tex]\[
15 \cdot (2)^{\frac{1}{3}}
\][/tex]
5. Compare with Given Choices:
[tex]\[
\boxed{15 \cdot \sqrt[3]{2}}
\][/tex]
We see that this matches one of the given choices.
Therefore, the correct answer is:
[tex]\[
15 \cdot \sqrt[3]{2}
\][/tex]
Thus, the answer is [tex]\(\boxed{1}\)[/tex].
