Alright, let's simplify the given mathematical expression step-by-step:
Given expression:
[tex]\[
\frac{(z^2)^5}{(z^7)^8}
\][/tex]
1. Simplify the numerator:
- The numerator is [tex]\((z^2)^5\)[/tex].
- Using the exponentiation rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we can simplify [tex]\((z^2)^5\)[/tex] as follows:
[tex]\[
(z^2)^5 = z^{2 \cdot 5} = z^{10}
\][/tex]
2. Simplify the denominator:
- The denominator is [tex]\((z^7)^8\)[/tex].
- Using the same exponentiation rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex], we simplify [tex]\((z^7)^8\)[/tex] as follows:
[tex]\[
(z^7)^8 = z^{7 \cdot 8} = z^{56}
\][/tex]
3. Combine the simplified numerator and denominator:
- We now have the expression:
[tex]\[
\frac{z^{10}}{z^{56}}
\][/tex]
- Using the rule [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we can subtract the exponents:
[tex]\[
\frac{z^{10}}{z^{56}} = z^{10-56} = z^{-46}
\][/tex]
So the final simplified expression is:
[tex]\[
\frac{\left(z^2\right)^5}{\left(z^7\right)^8} = z^{-46}
\][/tex]
