Question

1. \\(\frac{2a^{-3}b^{5}}{(a^{-5}b^{5})^{-2}}\\)

Explanation:

Step1: Simplify denominator exponent

Apply power rule $(x^m)^n=x^{mn}$
$(a^{-5}b^{5})^{-2}=a^{(-5)\times(-2)}b^{5\times(-2)}=a^{10}b^{-10}$

Step2: Rewrite the original expression

Substitute simplified denominator
$\frac{2a^{-3}b^{5}}{a^{10}b^{-10}}$

Step3: Combine like terms via exponent rules

Use $\frac{x^m}{x^n}=x^{m-n}$
$2a^{-3-10}b^{5-(-10)}=2a^{-13}b^{15}$

Step4: Rewrite positive exponents (optional)

Convert negative exponent to fraction
$2\times\frac{b^{15}}{a^{13}}=\frac{2b^{15}}{a^{13}}$

Answer:

$\frac{2b^{15}}{a^{13}}$ or $2a^{-13}b^{15}$