To determine the exponent of the variable [tex]\( b \)[/tex] in the simplified expression [tex]\(\frac{-4 a^{-2} b^4}{8 a^{-6} b^{-3}}\)[/tex], we need to follow a systematic approach to simplify the given expression step-by-step.
Let's break it down:
1. Simplify the constants:
- In the fraction [tex]\(\frac{-4}{8}\)[/tex], simplify -4 divided by 8:
[tex]\[
\frac{-4}{8} = -\frac{1}{2}
\][/tex]
2. Simplify the exponents for [tex]\( a \)[/tex]:
- In the expression, we have [tex]\( a^{-2} \)[/tex] in the numerator and [tex]\( a^{-6} \)[/tex] in the denominator. Using the property of exponents [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:
[tex]\[
\frac{a^{-2}}{a^{-6}} = a^{-2 - (-6)} = a^{-2 + 6} = a^{4}
\][/tex]
3. Simplify the exponents for [tex]\( b \)[/tex]:
- In the expression, we have [tex]\( b^{4} \)[/tex] in the numerator and [tex]\( b^{-3} \)[/tex] in the denominator. Again, using the property of exponents [tex]\(\frac{b^m}{b^n} = b^{m-n}\)[/tex]:
[tex]\[
\frac{b^4}{b^{-3}} = b^{4 - (-3)} = b^{4 + 3} = b^{7}
\][/tex]
So, putting all the simplified parts together, Marina's simplified expression is:
[tex]\[
-\frac{1}{2} a^4 b^{7}
\][/tex]
Thus, the exponent of the variable [tex]\( b \)[/tex] in Marina's solution is [tex]\( 7 \)[/tex].
Therefore, the box ( [tex]\(\square\)[/tex] ) should be filled with the number:
[tex]\[
\boxed{7}
\][/tex]
