Certainly! Let's simplify each of the given mathematical expressions step by step.
### Expression 1:
[tex]\[
\frac{1}{z^{-3}}
\][/tex]
To simplify this, recall that [tex]\(z^{-3}\)[/tex] can be rewritten as [tex]\(\frac{1}{z^3}\)[/tex]. Applying this, we get:
[tex]\[
\frac{1}{z^{-3}} = z^3
\][/tex]
So, the simplified form of [tex]\(\frac{1}{z^{-3}}\)[/tex] is [tex]\(z^3\)[/tex].
### Expression 2:
[tex]\[
-3z
\][/tex]
This expression is already in its simplest form. There are no further simplifications needed. Hence, the simplified form is:
[tex]\[
-3z
\][/tex]
### Expression 3:
[tex]\[
\frac{1}{-z^3}
\][/tex]
To simplify this, recognize that a negative exponent retains its effect:
Since [tex]\(-z^3\)[/tex] remains in the denominator, the negative sign can be factored out as follows,
[tex]\[
\frac{1}{-z^3} = -\frac{1}{z^3}
\][/tex]
This is the simplified form of the expression.
### Expression 4:
[tex]\[
z^3
\][/tex]
This expression is already in its simplest form. Therefore, the simplified form is:
[tex]\[
z^3
\][/tex]
Putting it all together, the simplified forms of the given expressions are:
1. [tex]\(\frac{1}{z^{-3}}\)[/tex] simplifies to [tex]\(z^3\)[/tex].
2. [tex]\(-3z\)[/tex] stays [tex]\(-3z\)[/tex].
3. [tex]\(\frac{1}{-z^3}\)[/tex] simplifies to [tex]\(-\frac{1}{z^3}\)[/tex].
4. [tex]\(z^3\)[/tex] stays [tex]\(z^3\)[/tex].
Thus, the final results are:
[tex]\[
(z^3, -3z, -1/z^3, z^3)
\][/tex]
