Question

which expression is equivalent to \\(\sqrt3{200k^{15}}\\), if \\(k \
eq 0\\)?\
\\(\bigcirc\\) a. \\(2k^{12}\sqrt3{25}\\)\
\\(\bigcirc\\) b. \\(2k^{5}\sqrt3{25}\\)\
\\(\bigcirc\\) c. \\(8k^{5}\sqrt3{25}\\)\
\\(\bigcirc\\) d. \\(8k\sqrt3{25}\\)

Explanation:

Step1: Factor the radicand

We can rewrite \(200k^{15}\) as a product of a perfect cube and another factor. First, factor \(200\): \(200 = 8\times25\), where \(8 = 2^3\) is a perfect cube. For the variable part, \(k^{15}=(k^{5})^3\) since \((a^m)^n=a^{mn}\) and \(5\times3 = 15\). So, \(\sqrt[3]{200k^{15}}=\sqrt[3]{8\times25\times(k^{5})^3}\).

Step2: Use the property of cube roots

The property of cube roots states that \(\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}\) for real numbers \(a\) and \(b\). Applying this property, we get:

$$
LATEXBLOCK0
$$

Because \(\sqrt[3]{8} = 2\) (since \(2^3=8\)) and \(\sqrt[3]{(k^{5})^3}=k^{5}\) (by the property \(\sqrt[n]{a^n}=a\) for odd \(n\)).

Answer:

B. \(2k^{5}\sqrt[3]{25}\)