Question

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

Explanation:

Step1: Apply power of a product rule

$(ab)^n=a^n b^n$, so:
$$(4x^{-4}y^3)^{-3}=4^{-3} \cdot (x^{-4})^{-3} \cdot (y^3)^{-3}$$

Step2: Apply power of a power rule

$(a^m)^n=a^{m \cdot n}$, so:
$$4^{-3} \cdot x^{(-4)(-3)} \cdot y^{(3)(-3)} = 4^{-3}x^{12}y^{-9}$$

Step3: Convert to positive exponents

$a^{-n}=\frac{1}{a^n}$, so:
$$4^{-3}=\frac{1}{4^3}=\frac{1}{64}, \quad y^{-9}=\frac{1}{y^9}$$
Substitute these in:
$$\frac{1}{64}x^{12} \cdot \frac{1}{y^9}$$

Step4: Combine terms

Multiply the terms together:
$$\frac{x^{12}}{64y^9}$$

Answer:

$\frac{x^{12}}{64y^9}$