Question

1. $y^{3} cdot (x^{-3} y^{-2})^{4}$

Explanation:

Step1: Apply Power of a Product Rule

For \((x^{-3}y^{-2})^4\), use \((ab)^n = a^n b^n\) and \((a^m)^n = a^{mn}\).
So, \((x^{-3})^4 = x^{-3\times4} = x^{-12}\) and \((y^{-2})^4 = y^{-2\times4} = y^{-8}\).
Now the expression is \(y^3 \cdot x^{-12} y^{-8}\).

Step2: Apply Product of Powers Rule

For \(y^3 \cdot y^{-8}\), use \(a^m \cdot a^n = a^{m + n}\).
\(y^{3 + (-8)} = y^{-5}\).
The expression becomes \(x^{-12} y^{-5}\), or \(\frac{1}{x^{12} y^5}\) (using \(a^{-n} = \frac{1}{a^n}\)).

Answer:

\(\frac{y^3}{x^{12} y^8}\) (simplified to \(\frac{1}{x^{12} y^5}\) or \(x^{-12} y^{-5}\))