Question

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

Explanation:

Step1: Apply power of a product rule

$(ab)^n=a^nb^n$, so:
$$(2y^{-4}z^5)^{-3}=2^{-3} \cdot (y^{-4})^{-3} \cdot (z^5)^{-3}$$

Step2: Simplify each exponent term

Use $(a^m)^n=a^{mn}$ and $a^{-n}=\frac{1}{a^n}$:
$$2^{-3}=\frac{1}{2^3}=\frac{1}{8}, \quad (y^{-4})^{-3}=y^{(-4)(-3)}=y^{12}, \quad (z^5)^{-3}=z^{(5)(-3)}=z^{-15}=\frac{1}{z^{15}}$$

Step3: Combine all simplified terms

Multiply the results together:
$$\frac{1}{8} \cdot y^{12} \cdot \frac{1}{z^{15}} = \frac{y^{12}}{8z^{15}}$$

Answer:

$\frac{y^{12}}{8z^{15}}$