Answer:
[tex]\Large\text{$4^{3}$}[/tex]
Step-by-step explanation:
Given expression:
[tex]\left(3^{-2}\cdot 4^{-5}\cdot 5^0\right)^{-3}\cdot \left(\dfrac{4^{-4}}{3^3}\right)^3\cdot 3^3[/tex]
To simplify the given expression, we can use the laws of exponents:
[tex]\boxed{\begin{array}{c}\underline{\textsf{Laws of Exponents}}\\\\\textsf{Product:}\;\;a^m \times a^n=a^{m+n}\\\\\textsf{Power of a Power:}\;\;(a^m)^n=a^{mn}\\\\\textsf{Power of a Quotient:}\;\;\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}\\\\\textsf{Power of a Product:}\;\;(ab)^m=a^mb^m\\\\\textsf{Zero Exponent:}\;\;a^0=1\end{array}}[/tex]
Begin by applying the power of a product rule and the power of a quotient rule:
[tex]\left(3^{-2}\right)^{-3}\cdot \left(4^{-5}\right)^{-3}\cdot \left(5^0\right)^{-3}\cdot \left(\dfrac{\left(4^{-4}\right)^3}{\left(3^3\right)^3}\right)\cdot 3^3[/tex]
Now, apply the power of a power rule:
[tex]3^{(-2 \cdot -3)}\cdot 4^{(-5 \cdot -3)}\cdot 5^{(0 \cdot -3)}\cdot \dfrac{4^{(-4\cdot 3)}}{3^{(3\cdot3)}}\cdot 3^3[/tex]
[tex]3^{6}\cdot 4^{15}\cdot 5^{0}\cdot \dfrac{4^{-12}}{3^{9}}\cdot 3^3[/tex]
Apply the zero exponent rule:
[tex]3^{6}\cdot 4^{15}\cdot 1\cdot \dfrac{4^{-12}}{3^{9}}\cdot 3^3[/tex]
[tex]3^{6}\cdot 4^{15}\cdot \dfrac{4^{-12}}{3^{9}}\cdot 3^3[/tex]
Collect like terms:
[tex]\dfrac{3^{6}\cdot 3^3}{3^{9}}\cdot 4^{15}\cdot 4^{-12}[/tex]
Apply the product rule:
[tex]\dfrac{3^{6+3}}{3^{9}}\cdot 4^{15-12}[/tex]
[tex]\dfrac{3^{9}}{3^{9}}\cdot 4^{3}[/tex]
[tex]1\cdot 4^{3}[/tex]
[tex]4^{3}[/tex]
Therefore, the simplified expression is:
[tex]\LARGE\boxed{\boxed{4^3}}[/tex]
