Sure! Let's go through the process of simplifying the given expression step by step.
We start with the expression:
[tex]\[
\left(\frac{a^2 b^{-5} c^3}{a^{-2} c^{-4}}\right)^4
\][/tex]
First, we simplify the fraction inside the parentheses. We do this by combining the exponents for each variable. Using the properties of exponents, we have:
[tex]\[
\frac{a^2 b^{-5} c^3}{a^{-2} c^{-4}} = a^{2 - (-2)} \cdot b^{-5} \cdot c^{3 - (-4)}
\][/tex]
Simplify the exponents:
[tex]\[
a^{2 - (-2)} = a^{2 + 2} = a^4
\][/tex]
[tex]\[
c^{3 - (-4)} = c^{3 + 4} = c^7
\][/tex]
Thus, the expression inside the fraction becomes:
[tex]\[
a^4 \cdot b^{-5} \cdot c^7 = \frac{a^4 c^7}{b^5}
\][/tex]
So the given expression now looks like:
[tex]\[
\left(\frac{a^4 c^7}{b^5}\right)^4
\][/tex]
Next, we apply the exponent 4 to each term inside the parentheses:
[tex]\[
\left(\frac{a^4 c^7}{b^5}\right)^4 = \frac{(a^4)^4 (c^7)^4}{(b^5)^4}
\][/tex]
Simplify the exponents:
[tex]\[
(a^4)^4 = a^{4 \cdot 4} = a^{16}
\][/tex]
[tex]\[
(c^7)^4 = c^{7 \cdot 4} = c^{28}
\][/tex]
[tex]\[
(b^5)^4 = b^{5 \cdot 4} = b^{20}
\][/tex]
So the expression becomes:
[tex]\[
\frac{a^{16} c^{28}}{b^{20}}
\][/tex]
Therefore, the simplified form of the expression using only positive exponents is:
[tex]\[
\boxed{\frac{a^{16} c^{28}}{b^{20}}}
\][/tex]
