Question

simplify.
$(2x^{-5}y^{6})^{5}$
write your answer using only positive exponents.

Explanation:

Step1: Apply power of a product rule

Distribute the exponent 5 to each term inside the parentheses using the rule $(ab)^n=a^n b^n$:
$$(2x^{-5}y^6)^5 = 2^5 \cdot (x^{-5})^5 \cdot (y^6)^5$$

Step2: Simplify each term's exponent

Use the power rule $(a^m)^n=a^{m \cdot n}$ and calculate $2^5$:
$$2^5=32,\quad (x^{-5})^5=x^{-5 \cdot 5}=x^{-25},\quad (y^6)^5=y^{6 \cdot 5}=y^{30}$$
Combining these gives:
$$32x^{-25}y^{30}$$

Step3: Convert negative exponent to positive

Use the rule $a^{-n}=\frac{1}{a^n}$ to rewrite $x^{-25}$ with a positive exponent:
$$32 \cdot \frac{1}{x^{25}} \cdot y^{30} = \frac{32y^{30}}{x^{25}}$$

Answer:

$\frac{32y^{30}}{x^{25}}$