Option C:
[tex]\frac{3 a^{2} b^{11}}{2 }[/tex] is equivalent to the given expression.
Solution:
Given expression:
[tex]$\frac{-18 a^{-2} b^{5}}{-12 a^{-4} b^{-6}}[/tex]
To find which expression is equivalent to the given expression.
[tex]$\frac{-18 a^{-2} b^{5}}{-12 a^{-4} b^{-6}}[/tex]
Using exponent rule: [tex]\frac{1}{a^m}=a^{-m}, \ \ \frac{1}{a^{-m}}=a^{m}[/tex]
[tex]$=\frac{-18 a^{-2} b^{5}a^{4} b^{6}}{-12 }[/tex]
[tex]$=\frac{-18 a^{-2} a^{4} b^{5} b^{6}}{-12 }[/tex]
Using exponent rule: [tex]{a^m}\cdot{a^n}=a^{m+n}[/tex]
[tex]$=\frac{-18 a^{(-2+4)} b^{(5+6)}}{-12 }[/tex]
[tex]$=\frac{-18 a^{2} b^{11}}{-12 }[/tex]
Divide both numerator and denominator by the common factor –6.
[tex]$=\frac{3 a^{2} b^{11}}{2 }[/tex]
[tex]$\frac{-18 a^{-2} b^{5}}{-12 a^{-4} b^{-6}}=\frac{3 a^{2} b^{11}}{2 }[/tex]
Therefore, [tex]\frac{3 a^{2} b^{11}}{2 }[/tex] is equivalent to the given expression.
Hence Option C is the correct answer.
