To evaluate the given expression, we use PEMDAS.

First, we compute the exponents:

\begin{align*} \left(-3\right)^3+4^3= \left(-27\right)+64 \end{align*}

Then, we add:

\begin{align*} \left(-27\right)+64 &= \\[5pt] -27+64 &= \\[5pt] 37 \end{align*}

To evaluate the given expression, we use PEMDAS.

First, we compute the exponent:

\begin{align*} 4 \times \left(-2\right)^4= 4 \times 16 \end{align*}

Then, we multiply:

\begin{align*} 4 \times 16&= \bbox[3pt,Gainsboro]{\color{blue}64} \end{align*}

To evaluate the given expression, we use PEMDAS.

First, we compute the exponent:

\begin{align*} 6\div 3 + \left(-3\right)^3 &= 6\div 3 - 27 \end{align*}

Next, we divide:

\begin{align*} 6\div 3 -27 &= 2 - 27 \end{align*}

Finally, we subtract: \begin{align} 2 -27 &= \bbox[3pt,Gainsboro]{\color{blue}-25} \end{align}

To evaluate this expression, we use PEMDAS.

First, we perform the operation inside the parentheses:

\begin{align*} \left(1-(-3)\right)^3&= \left(4\right)^3 \end{align*}

Next, we compute the exponent:

\begin{align*} \left(4\right)^3 &= \bbox[3pt,Gainsboro]{\color{blue}64} \end{align*}

To evaluate this expression, we use PEMDAS.

First, we perform the operation inside the parentheses:

\begin{align*} \left(-1-2\right)^3&= \left(-3\right)^3 \end{align*}

Next, we compute the exponent:

\begin{align*} \left(-3\right)^3 &= -27 \end{align*}

To evaluate this expression, we use PEMDAS.

We begin by performing the operations inside the parentheses. So, first we divide $4$ by $(-2)$:

\begin{align*} (4\div (-2)+5)^4&= ((-2)+5)^4 \end{align*}

Then, we add the numbers inside the parentheses:

\begin{align*} ((-2)+5)^4&= 3^4 \end{align*}

Finally, we compute the exponent:

\begin{align*} 3^4=\bbox[3pt,Gainsboro]{\color{blue}81} \end{align*}

To evaluate this expression, we use PEMDAS.

First, consider the expression inside the parentheses: \begin{align*} (-3)^2+ (-2)^3 \end{align*}

Applying PEMDAS to this expression, we obtain: \begin{align*} (-3)^2+ (-2)^4 &= \\[5pt] 9 + 16 &= \\[5pt] 25 \end{align*}

Therefore: \begin{align*} \require{cancel} \left( (-3)^2+ (-2)^4 \right) \times\left( -\dfrac15 \right)&= \\[4pt] 25 \times \left( -\dfrac15 \right) &= \\[5pt] \dfrac{25}{1}\times \left( -\dfrac15 \right)&= \\[5pt] -\left(\dfrac{25}{1}\times \dfrac15\right)&= \\[5pt] -\dfrac{25\times 1}{1\times 5}&= \\[5pt] -\dfrac{25}{ 5}&= \\[5pt] \bbox[3pt,Gainsboro]{\color{blue}-5} \end{align*}

To evaluate this expression, we use PEMDAS.

First, consider the expression inside the parentheses:

\begin{align*} (-4)^3 \div 8 \end{align*}

Applying PEMDAS to this expression, we obtain:

\begin{align*} (-4)^3 \div 8 &= \\[5pt] (-64) \div 8 &= \\[5pt] \dfrac{(-64)}{8} &= \\[5pt] -8 \end{align*}

Therefore:

\begin{align*} \require{cancel} \left(-\dfrac 1 3\right)\div \left( (-4)^3 \div 8 \right) &= \\[4pt] \left(-\dfrac 1 3\right) \div (-8) &= \\[5pt] \left(-\dfrac{1}{3}\right) \times \left(-\dfrac 1 8\right) &= \\[5pt] \dfrac{1\times 1}{3\times 8} &= \\[5pt] \dfrac{1}{24} \end{align*}

To evaluate this expression, we use PEMDAS.

First, we consider the expressions inside each of the parentheses. The first one is \begin{align*} (-4)^2+5\times2. \end{align*}

Applying PEMDAS to this expression, we obtain: \begin{align*} (-4)^2+5 \times 2 &= \\[5pt] 16+5\times2 &= \\[5pt] 16+10 &= \\[5pt] 26 \end{align*}

The expression inside the second parentheses is \begin{align*} (-2)^3\div(-4). \end{align*}

Applying PEMDAS to this expression, we obtain: \begin{align*} (-2)^3\div(-4) &= \\[5pt] (-8) \div (-4) &= \\[5pt] \dfrac{(-8)}{(-4)} &= \\[5pt] %\dfrac84&= \\[5pt] 2 \end{align*}

Therefore: \begin{align*} \require{cancel} \require{cancel} \left((-4)^2+5\times2\right)-\left((-2)^3\div(-4)\right) &= \\[5pt] 26-2&=\\[5pt] \bbox[3pt,Gainsboro]{\color{blue}24} \end{align*}