Let's go through the process of simplifying the given expression [tex]\(\frac{4 b}{a^{-10}}\)[/tex].
1. Identify the negative exponent: The term [tex]\(a^{-10}\)[/tex] in the denominator has a negative exponent, which we can handle using the property of exponents that states [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex].
2. Apply the property of exponents: According to the property, [tex]\(a^{-10}\)[/tex] can be rewritten as [tex]\(\frac{1}{a^{10}}\)[/tex].
3. Rewrite the expression using the property: Now, substitute [tex]\(\frac{1}{a^{10}}\)[/tex] back into the original expression:
[tex]\[
\frac{4 b}{a^{-10}} = 4 b \times \frac{1}{a^{-(-10)}} = 4 b \times a^{10}
\][/tex]
4. Simplify the expression: As multiplication is associative and commutative, we can write the expression as:
[tex]\[
4 b \times a^{10} = 4 a^{10} b
\][/tex]
Thus, the correct simplification of the expression [tex]\(\frac{4 b}{a^{-10}}\)[/tex] is:
[tex]\[
4 a^{10} b
\][/tex]
Therefore, the correct choice is [tex]\(4 a^{10} b\)[/tex].
