Question

simplify and write answer without negative exponents.
$(x^{3}y^{-2})^{-2}(xy^{-3})^{-3}$
\bigcirc $x^{18}y^{36}$
\bigcirc $\frac{y^{13}}{x^{9}}$
\bigcirc $x^{9}y^{13}$
\bigcirc $\frac{y^{36}}{x^{18}}$

Explanation:

Step1: Apply Power of a Product Rule

For \((x^{3}y^{-2})^{-2}\), use \((ab)^n = a^n b^n\) and \((a^m)^n = a^{mn}\).
\(x^{3\times(-2)}y^{-2\times(-2)} = x^{-6}y^{4}\).

For \((xy^{-3})^{-3}\), apply the same rules:
\(x^{1\times(-3)}y^{-3\times(-3)} = x^{-3}y^{9}\).

Step2: Multiply the Two Expressions

Multiply \(x^{-6}y^{4}\) and \(x^{-3}y^{9}\) using \(a^m \cdot a^n = a^{m + n}\).
For \(x\): \(x^{-6} \cdot x^{-3} = x^{-6 + (-3)} = x^{-9}\).
For \(y\): \(y^{4} \cdot y^{9} = y^{4 + 9} = y^{13}\).
So we have \(x^{-9}y^{13}\).

Step3: Eliminate Negative Exponent

Use \(a^{-n} = \frac{1}{a^n}\) for \(x^{-9}\):
\(x^{-9}y^{13} = \frac{y^{13}}{x^{9}}\).

Answer:

\(\frac{y^{13}}{x^{9}}\) (corresponding to the option " \(\boldsymbol{\frac{y^{13}}{x^{9}}}\) ")