Question

simplify.
$\frac{(x^{-5}y^{4})^{-2}}{(3x^{5}y^{-4})^{3}}$
write your answer using only positive exponents.

Explanation:

Step1: Apply power of a product rule

For numerator: $(x^{-5}y^{4})^{-2} = x^{(-5)\times(-2)}y^{4\times(-2)} = x^{10}y^{-8}$
For denominator: $(3x^{5}y^{-4})^{3} = 3^{3}x^{5\times3}y^{(-4)\times3} = 27x^{15}y^{-12}$
Now the expression becomes: $\frac{x^{10}y^{-8}}{27x^{15}y^{-12}}$

Step2: Apply quotient of powers rule

For $x$-terms: $\frac{x^{10}}{x^{15}} = x^{10-15} = x^{-5}$
For $y$-terms: $\frac{y^{-8}}{y^{-12}} = y^{-8-(-12)} = y^{4}$
Now the expression becomes: $\frac{x^{-5}y^{4}}{27}$

Step3: Convert to positive exponents

$x^{-5} = \frac{1}{x^{5}}$, so substitute: $\frac{y^{4}}{27x^{5}}$

Answer:

$\frac{y^{4}}{27x^{5}}$