- Factor the constant 200 into \(2^3 \cdot 25\).
- Separate the expression into individual radicals: \(\sqrt[3]{2^3} \cdot \sqrt[3]{25} \cdot \sqrt[3]{k^{15}}\).
- Simplify each part to obtain \(2 \cdot \sqrt[3]{25} \cdot k^5\).
- Reorder the terms to match the correct format: \(\boxed{2 k^5 \sqrt[3]{25}}\).
Explanation
- Factoring the Expression
To simplify the radical expression \(\sqrt[3]{200 k^{15}}\) for \(k
eq 0\), we first factor the constant term under the cube root. The number 200 can be factored into prime powers as \(200 = 8 \times 25 = 2^3 \times 5^2\). Substituting this into the original expression, we have:
\[\sqrt[3]{200 k^{15}} = \sqrt[3]{2^3 \times 25 \times k^{15}}.\]
- Separating the Cube Roots
Next, we apply the property of radicals that states \(\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}\) to separate the cube root of the product into the product of the cube roots of each factor:
\[\sqrt[3]{2^3 \times 25 \times k^{15}} = \sqrt[3]{2^3} \cdot \sqrt[3]{25} \cdot \sqrt[3]{k^{15}}.\]
- Simplifying and Combining Terms
Now, we simplify each factor individually:
- For the constant factor, \(\sqrt[3]{2^3} = 2\).
- The term \(\sqrt[3]{25}\) remains as it is, since 25 is not a perfect cube.
- For the variable factor, we use the property \(\sqrt[n]{x^m} = x^{m/n}\), which gives \(\sqrt[3]{k^{15}} = k^{15/3} = k^5\).
Combining these simplified factors together, we get:
\[2 \cdot \sqrt[3]{25} \cdot k^5 = 2 k^5 \sqrt[3]{25}.\]
- Conclusion
Comparing our result \(2 k^5 \sqrt[3]{25}\) with the given options, we can see that it corresponds to the second option.
The correct answer is therefore \(\boxed{2 k^5 \sqrt[3]{25}}\).
Examples
Understanding how to simplify radical expressions is essential in engineering and physical sciences. For instance, when calculating the volume of a sphere given a variable formula related to material properties, one often encounters complex radicals. Simplification, as demonstrated here, allows researchers to calculate values efficiently and maintain accuracy in formulas used for structural stress analysis or heat distribution models.