Which expression is equivalent to [tex]\frac{-18 a^{-2} b^5}{-12 a^{-4} b^{-6}}[/tex]? Assume [tex]a \neq 0[/tex], [tex]b \neq 0[/tex].

A. [tex]\frac{2 a^2 b^{11}}{3}[/tex]
B. [tex]\frac{2 a^2 b^{30}}{3}[/tex]
C. [tex]\frac{3 a^2 b^{11}}{2}[/tex]
D. [tex]\frac{3 a^2 b^{30}}{2}[/tex]



Answer :

To find the equivalent expression for [tex]\(\frac{-18 a^{-2} b^5}{-12 a^{-4} b^{-6}}\)[/tex], we need to simplify the fraction carefully, following algebraic rules and properties of exponents.

1. Simplify the constants:
[tex]\[ \frac{-18}{-12} \][/tex]
The negative signs cancel out, simplifying to:
[tex]\[ \frac{18}{12} \][/tex]
Further simplification of the fraction gives:
[tex]\[ \frac{18 \div 6}{12 \div 6} = \frac{3}{2} \][/tex]

2. Simplify the exponents for [tex]\(a\)[/tex]:
[tex]\[ \frac{a^{-2}}{a^{-4}} \][/tex]
Using the properties of exponents, specifically [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we get:
[tex]\[ a^{-2 - (-4)} = a^{-2 + 4} = a^2 \][/tex]

3. Simplify the exponents for [tex]\(b\)[/tex]:
[tex]\[ \frac{b^5}{b^{-6}} \][/tex]
Using the same exponent rule:
[tex]\[ b^{5 - (-6)} = b^{5 + 6} = b^{11} \][/tex]

4. Combine the simplified components:
Putting it all together, we have:
[tex]\[ \frac{3}{2} \cdot a^2 \cdot b^{11} \][/tex]

Thus, the expression [tex]\(\frac{-18 a^{-2} b^5}{-12 a^{-4} b^{-6}}\)[/tex] simplifies to:

[tex]\[ \frac{3 a^2 b^{11}}{2} \][/tex]

The correct choice from the given options is:
[tex]\[ \boxed{\frac{3 a^2 b^{11}}{2}} \][/tex]