Question

1. $(-3p^{4}q^{-3})^{2}(4p^{-5}q^{7})$

Explanation:

Step1: Apply power of a product rule

First, we apply the power of a product rule \((ab)^n = a^n b^n\) and the power of a power rule \((a^m)^n = a^{mn}\) to \((-3p^4q^{-3})^2\).

\[

$$\begin{align*} (-3p^4q^{-3})^2&=(-3)^2(p^4)^2(q^{-3})^2\\ &= 9p^{8}q^{-6} \end{align*}$$

\]

Step2: Multiply with the second term

Now we multiply \(9p^{8}q^{-6}\) with \(4p^{-5}q^{7}\). When multiplying terms with the same base, we add the exponents (\(a^m \cdot a^n = a^{m + n}\)) and multiply the coefficients.

\[

$$\begin{align*} (9p^{8}q^{-6})(4p^{-5}q^{7})&=(9\times4)(p^{8}\cdot p^{-5})(q^{-6}\cdot q^{7})\\ &= 36p^{8 + (-5)}q^{-6 + 7}\\ &= 36p^{3}q^{1}\\ &= 36p^{3}q \end{align*}$$

\]

Answer:

\(36p^{3}q\)