Question

simplify the radical. assume that all variables represent positive real numbers.
\\(\sqrt5{-243x^{10}y^{5}}\\)

select the correct choice below and, if necessary, fill in the answer box to complete your choi
\\(\bigcirc\\) a. \\(\sqrt5{-243x^{10}y^{5}} = \square\\)
\\(\bigcirc\\) b. the radical does not represent a real number.

Explanation:

Step 1: Analyze the fifth root of -243

Recall that for odd roots, we can have negative radicands. We know that \((-3)^5=-243\), so \(\sqrt[5]{-243}=-3\).

Step 2: Analyze the fifth root of \(x^{10}\)

Using the property of radicals \(\sqrt[n]{a^m}=a^{\frac{m}{n}}\), for \(\sqrt[5]{x^{10}}\), we have \(x^{\frac{10}{5}} = x^{2}\).

Step 3: Analyze the fifth root of \(y^{5}\)

Using the property of radicals \(\sqrt[n]{a^n}=a\) (for odd \(n\) and real \(a\)), for \(\sqrt[5]{y^{5}}\), we get \(y\) (since \(y\) is a positive real number).

Step 4: Combine the results

Using the property \(\sqrt[n]{abc}=\sqrt[n]{a}\cdot\sqrt[n]{b}\cdot\sqrt[n]{c}\), we have:

$$
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Answer:

A. \(\boldsymbol{\sqrt[5]{-243x^{10}y^{5}}=-3x^{2}y}\)