Question

simplify.
$\frac{(3a^{-1}b^{-2})^{2}}{(a^{-3}b^{-4})^{-3}}$
write your answer using only positive exponents.

Explanation:

Step1: Apply power of a product rule

Raise each factor in numerator/denominator to the outer exponent.
Numerator: $(3a^{-1}b^{-2})^2 = 3^2 \cdot (a^{-1})^2 \cdot (b^{-2})^2 = 9a^{-2}b^{-4}$
Denominator: $(a^{-3}b^{-4})^{-3} = (a^{-3})^{-3} \cdot (b^{-4})^{-3} = a^{9}b^{12}$
Now the expression is: $\frac{9a^{-2}b^{-4}}{a^{9}b^{12}}$

Step2: Apply quotient of powers rule

Subtract exponents for like bases.
For $a$: $a^{-2-9}=a^{-11}$
For $b$: $b^{-4-12}=b^{-16}$
Expression becomes: $9a^{-11}b^{-16}$

Step3: Convert to positive exponents

Use $x^{-n}=\frac{1}{x^n}$ to rewrite negative exponents.
$9a^{-11}b^{-16} = \frac{9}{a^{11}b^{16}}$

Answer:

$\frac{9}{a^{11}b^{16}}$