Question

1. $2x^{2}y^{-6}cdot (x^{3}y^{5})^{-2}$

Explanation:

Step1: Expand the raised power

Apply exponent rule $(a^m b^n)^p = a^{m \cdot p} b^{n \cdot p}$
$(x^3 y^5)^{-2} = x^{3 \cdot (-2)} y^{5 \cdot (-2)} = x^{-6} y^{-10}$

Step2: Multiply like terms

Combine $2x^2 y^{-6}$ with $x^{-6} y^{-10}$, use $a^m \cdot a^n = a^{m+n}$
$2x^{2 + (-6)} y^{-6 + (-10)} = 2x^{-4} y^{-16}$

Step3: Rewrite with positive exponents

Use $a^{-n} = \frac{1}{a^n}$
$2 \cdot \frac{1}{x^4} \cdot \frac{1}{y^{16}} = \frac{2}{x^4 y^{16}}$

Answer:

$\frac{2}{x^4 y^{16}}$