For the following two (02) items:
Let $\alpha$ and $\beta $ be the roots of the quadratic equation
$(x^{2}+(\log _{0.5}(\alpha ^{2}))x+(\log _{0.5}(\alpha ^{2}))^{4}=0$
where $a ^ 2 \ne1$ and $\log _{0.5}(\alpha ^{2})>0$ Further, $\beta ^{2}=\alpha (\log _{\alpha ^{2}}(0.5))$
 

What is ẞ equal to?

For the following two (02) items:
Let $p=\sum_{j=1}^{n}\log_{10}2^{j} and q=\sum_{j=1}^{n}\log_{10}5^{j}$

If p + q = 66 then which one of the following is correct?

For the following two (02) items:
Let $p=\sum_{j=1}^{n}\log_{10}2^{j} and q=\sum_{j=1}^{n}\log_{10}5^{j}$

If p + q = 15 then what is equal to? q - p

If $\log _b a=p, \log _d c=2 p$ and $\log _f e=3 p$, then what is $(a c e)^{\frac{1}{p}}$ equal to ?