Answer
$$\frac{1}{100a^{4}b^{12}c^{8}}$$
Work Step by Step
$$(\frac{20a^{-3}b^{4}c^{5}}{2a^{-5}b^{-2}c})^{-2}$$
Recall the power rule: $(a^{m})^{n}=a^{mn}$
Thus,
$=(\frac{20a^{-3}b^{4}c^{5}}{2a^{-5}b^{-2}c})^{-2}$
$=(\frac{10a^{-3}b^{4}c^{5}}{a^{-5}b^{-2}c})^{-2}$
$=\frac{10^{-2}a^{-3\cdot-2}b^{4\cdot-2}c^{5\cdot-2}}{a^{-5\cdot-2}b^{-2\cdot-2}c^{-2}}$
$=\frac{10^{-2}a^{6}b^{-8}c^{-10}}{a^{10}b^{4}c^{-2}}$
Recall the negative exponent rule: $a^{−n}=\frac{1}{a^{n}}$ and $\frac{1}{a^{-n}} = a^{n}$
Thus,
$=\frac{10^{-2}a^{6}b^{-8}c^{-10}}{a^{10}b^{4}c^{-2}}$
$=\frac{a^{6}b^{-8}c^{-10}}{10^{2}a^{10}b^{4}c^{-2}}$
Recall the quotient rule: $\frac{a^{m}}{a^{n}}=a^{m-n}$ and $\frac{a^{n}}{a^{m}}=\frac{1}{a^{m+n}}$ if $m>n$
$=\frac{a^{6}b^{-8}c^{-10}}{10^{2}a^{10}b^{4}c^{-2}}$
$=\frac{1}{10^{2}a^{10-6}b^{4+8}c^{-2+10}}$
$=\frac{1}{10^{2}a^{4}b^{12}c^{8}}$
$=\frac{1}{100a^{4}b^{12}c^{8}}$
