Answer
7
Work Step by Step
We use the Properties of Exponents, step by step...
P0. $a^{0}=1,\ a^{1}=a$
P1. $a^{m}\cdot a^{n}=a^{m+n}$
P2. $\displaystyle \frac{a^{m}}{a^{n}}=a^{m-n}$ , $ \displaystyle \frac{1}{a^{n}}=a^{-n}$
P3. $(a^{m})^{n}=a^{mn}$
P4. $(ab)^{m}=a^{m} b^{m}$
P5. $(\displaystyle \frac{a}{b})^{m}=\frac{a^{m}}{b^{m}}$
P6. Rational exponents: $a^{m/n}=(a^{1/n})^{m} = \sqrt[n]{a^{m}}$
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$(\displaystyle \frac{7^{-12}\cdot 7^{3}}{7^{-8}})^{-1}$= ... P1...
$=(\displaystyle \frac{7^{-12+3}}{7^{-8}})^{-1}=(\frac{7^{-9}}{7^{-8}})^{-1}$ = ...P2...
=$(7^{-9-(-8)})^{-1}=(7^{-1})^{-1}$ = ... P3...
=$ 7^{-1(-1)}=7^{1}$= ...P0...
= $7$
