Answer
$a)$ negative
$b)$ positive
$c)$ negative
$d)$ negative
$e)$ positive
$f)$ negative
Work Step by Step
$a\gt0$ $,$ $b\lt0$ and $c\lt0$
$a)$ $b^{5}$
$b^{5}$ is a $\textbf{negative}$ expression because a negative number raised to and odd power is also negative.
$b)$ $b^{10}$
$b^{10}$ is a $\textbf{positive}$ expression because a negative number raised to an even power is positive.
$c)$ $ab^{2}c^{3}$
$a$ is a positive number, $b^{2}$ is positive because a negative number raised to an even power is positive and $c^{3}$ is negative because a negative number raised to an odd power is negative.
Knowing that, the product is positive$\times$positive$\times$negative and the result is a negative expression. For this reason, $ab^{2}c^{3}$ is a $\textbf{negative}$ expression.
$d)$ $(b-a)^{3}$
Since the operation inside the parentheses yields a negative number, $(b-a)^{3}$ is a $\textbf{negative}$ expression because a negative number raised to an odd power is also negative.
$e)$ $(b-a)^{4}$
Since the operation inside the parentheses yields a negative number, $(b-a)^{4}$ is a $\textbf{positive}$ expression because a negative number raised to an even power is positive.
$f)$ $\dfrac{a^{3}c^{3}}{b^{6}c^{6}}$
$a^{3}$ is positive because a positive number raised to an odd power is also positive. $c^{3}$ is negative because a negative number raised to an odd power is also negative. $b^{6}$ and $c^{6}$ are positive because a negative number raised to an even power is positive
Since the operation $\dfrac{(+)(-)}{(+)(+)}=\dfrac{-}{+}=-$, then $\dfrac{a^{3}c^{3}}{b^{6}c^{6}}$ is a $\textbf{negative}$ expression
