Sure, let's simplify the expression step by step:
[tex]$
\frac{(z^2)^5}{(z^7)^8}.
$[/tex]
Step 1: Simplify the numerator.
The numerator is [tex]\((z^2)^5\)[/tex]. Using the properties of exponents, specifically the power of a power property [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we get:
[tex]$
(z^2)^5 = z^{2 \cdot 5} = z^{10}.
$[/tex]
So the numerator simplifies to [tex]\(z^{10}\)[/tex].
Step 2: Simplify the denominator.
The denominator is [tex]\((z^7)^8\)[/tex]. Again, using the power of a power property, we get:
[tex]$
(z^7)^8 = z^{7 \cdot 8} = z^{56}.
$[/tex]
So the denominator simplifies to [tex]\(z^{56}\)[/tex].
Step 3: Simplify the entire expression by dividing the simplified numerator by the simplified denominator.
We need to simplify the expression [tex]\(\frac{z^{10}}{z^{56}}\)[/tex]. Using the property of exponents for division [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we get:
[tex]$
\frac{z^{10}}{z^{56}} = z^{10-56} = z^{-46}.
$[/tex]
Therefore, the completely simplified expression is:
[tex]$
\frac{(z^2)^5}{(z^7)^8} = z^{-46}.
$[/tex]
