To simplify the given expression [tex]\(\frac{6 a ^2 b^{-2}}{8 a^{-3} b^3}\)[/tex], let's proceed step-by-step:
1. Simplify the coefficient:
[tex]\[
\frac{6}{8} = 0.75
\][/tex]
2. Simplify the exponents of [tex]\(a\)[/tex]:
Using the property of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex],
[tex]\[
\frac{a^2}{a^{-3}} = a^{2 - (-3)} = a^{2 + 3} = a^5
\][/tex]
3. Simplify the exponents of [tex]\(b\)[/tex]:
Again, using the property of exponents [tex]\(\frac{b^m}{b^n} = b^{m-n}\)[/tex],
[tex]\[
\frac{b^{-2}}{b^3} = b^{-2 - 3} = b^{-5}
\][/tex]
4. Combine all the simplified parts:
[tex]\[
0.75 \cdot a^5 \cdot b^{-5} = \frac{3}{4} \cdot a^5 \cdot b^{-5}
\][/tex]
Since [tex]\(b^{-5}\)[/tex] can also be written as [tex]\(\frac{1}{b^5}\)[/tex], the expression becomes:
[tex]\[
\frac{3 a^5}{4 b^5}
\][/tex]
So, the simplified form of the given expression is:
\[
\boxed{\frac{3 a^5}{4 b^5}}
\
