Let's simplify the given expression step-by-step:
The original expression is:
[tex]\[
\frac{4 b}{a^{-10}}
\][/tex]
We know the property of exponents that states [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex]. Applying this property, we can rewrite [tex]\( a^{-10} \)[/tex] as:
[tex]\[
a^{-10} = \frac{1}{a^{10}}
\][/tex]
Substituting this into the original expression, we get:
[tex]\[
\frac{4 b}{\frac{1}{a^{10}}}
\][/tex]
To simplify the division by a fraction, multiply by the reciprocal:
[tex]\[
\frac{4 b}{\frac{1}{a^{10}}} = 4 b \times a^{10}
\][/tex]
So, we have:
[tex]\[
4 b \times a^{10} = 4 a^{10} b
\][/tex]
Therefore, the correct simplification of the expression [tex]\(\frac{4 b}{a^{-10}}\)[/tex] is:
[tex]\[
4 a^{10} b
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{4 a^{10} b}
\][/tex]
