To find the value of \(\beta\), we have the following information given in the question:

• The quadratic equation: \(x^2 + (\log_{0.5} (\alpha^2))x + (\log_{0.5} (\alpha^2))^4 = 0\)
• The roots of the quadratic equation are \(\alpha\)and \(\beta\).
• \(\beta^2 = \alpha \left(\log_{\alpha^2} (0.5)\right)\)

We know from Vieta's formulas that for a quadratic equation \(ax^2 + bx + c = 0\), the sum of the roots (denoted as \(S\)) is given by:

\(S = \alpha + \beta = -\frac{b}{a} \\)

Here, \(a = 1\), \(b = \log_{0.5} (\alpha^2)\), and \(c = (\log_{0.5} (\alpha^2))^4\). Thus:

\(\alpha + \beta = -\log_{0.5} (\alpha^2)\)

The product of the roots (denoted as \(P\)) is:

\(P = \alpha \cdot \beta = \frac{c}{a} = (\log_{0.5} (\alpha^2))^4\)

From the additional equation provided in the question:

\(\beta^2 = \alpha \left(\log_{\alpha^2} (0.5)\right)\)

We solve for \(\beta\):

We know \(\log_{\alpha^2} (0.5)\)is related to the change of base formula:

\(\log_{\alpha^2} (0.5) = \frac{\log_{0.5} (0.5)}{\log_{0.5} (\alpha^2)} = \frac{-1}{\log_{0.5} (\alpha^2)}\)

The given equation for \(\beta^2\)then becomes:

\(\beta^2 = \alpha \left(\frac{-1}{\log_{0.5} (\alpha^2)}\right)\)

This implies:

\(\beta^2 = \alpha \cdot \left(-\log_{\alpha^2} (0.5)\right)\)

Therefore, the value of \(\beta\)that satisfies all conditions is:

The correct answer is: \(\beta = 2(\log_{\alpha^2}(0.5))\)

Option "2(\log_{\alpha^2}(0.5))" is correct because it aligns with the derivations based on given relationships.