To simplify and rewrite \((4x^2y^{-3})^{-2}\) using only positive exponents, follow these steps:
- Apply the negative exponent rule: \((a^{-n} = \frac{1}{a^{n}})\).
- Raise each part of the expression inside the parentheses to the power of -2.
Starting with the expression:
\[ (4x^2y^{-3})^{-2} \]
This can be rewritten as:
\[ \frac{1}{(4x^2y^{-3})^2} \]
Now, we calculate \((4x^2y^{-3})^2\):
\[ (4^2)(x^{2 \cdot 2})(y^{-3 \cdot 2}) = 16x^4y^{-6} \]
Now, substituting that back we get:
\[ \frac{1}{16x^4y^{-6}} \]
To eliminate the negative exponent, remember that \(y^{-6}\) in the denominator can be rewritten as \( \frac{1}{y^6} \):
\[ \frac{1}{16x^4} \cdot y^6 = \frac{y^6}{16x^4} \]
Thus, the simplified expression using only positive exponents is:
\[ \frac{y^6}{16x^4} \]
This is the final answer.