To simplify the expression [tex]\(\frac{-2 a^2 b^4}{4 a b^{-8}}\)[/tex] and express the answer using positive exponents, follow these steps:
1. Simplify the coefficients:
The coefficients in the numerator and denominator are [tex]\(-2\)[/tex] and [tex]\(4\)[/tex], respectively:
[tex]\[
\frac{-2}{4} = -\frac{1}{2}
\][/tex]
2. Simplify the exponents of [tex]\(a\)[/tex]:
The exponents of [tex]\(a\)[/tex] in the numerator and denominator are [tex]\(2\)[/tex] and [tex]\(1\)[/tex], respectively:
[tex]\[
\frac{a^2}{a} = a^{2-1} = a^1 = a
\][/tex]
3. Simplify the exponents of [tex]\(b\)[/tex]:
The exponents of [tex]\(b\)[/tex] in the numerator and denominator are [tex]\(4\)[/tex] and [tex]\(-8\)[/tex], respectively:
[tex]\[
\frac{b^4}{b^{-8}} = b^{4 - (-8)} = b^{4+8} = b^{12}
\][/tex]
4. Combine all the simplified parts:
Putting it all together, the simplified expression is:
[tex]\[
-\frac{1}{2} a b^{12}
\][/tex]
Thus, the simplified expression [tex]\(\frac{-2 a^2 b^4}{4 a b^{-8}}\)[/tex] is:
[tex]\[
-\frac{1}{2} a b^{12}
\][/tex]
From the given options, it is clear that:
[tex]\[
-\frac{1}{2} a b^{12}
\][/tex]
is the correct and simplified form of the given expression.
