Sure, let's simplify the expression \(\frac{z^{-8}}{z^{-7}}\) step by step and write the answer with a positive exponent.
1. Start with the given expression:
[tex]\[
\frac{z^{-8}}{z^{-7}}
\][/tex]
2. Apply the laws of exponents:
When dividing powers with the same base, subtract the exponent of the denominator from the exponent of the numerator:
[tex]\[
\frac{z^{-8}}{z^{-7}} = z^{-8 - (-7)}
\][/tex]
Simplify the exponent:
[tex]\[
-8 - (-7) = -8 + 7 = -1
\][/tex]
So the expression becomes:
[tex]\[
z^{-1}
\][/tex]
3. Convert the result to a positive exponent:
Recall that \(z^{-1} = \frac{1}{z^1}\). Therefore, we can rewrite:
[tex]\[
z^{-1} = \frac{1}{z^1}
\][/tex]
Putting it all together, the simplified expression with a positive exponent is:
[tex]\[
\frac{1}{z^1}
\][/tex]
So, the answer is:
[tex]\[
\frac{1}{z}
\][/tex]
