What order of operation should I use calculating $2^3 \times 4 - 6 \times 3 ÷ (4 + 3) - 16 ÷ 4$? I am struggling if I should use PEMDAS or BODMAS in this equation. What is the right method to get around these type of equations?
$$2^3 \times 4 - 6 \times 3 ÷ (4 + 3) - 16 ÷ 4 = \ ?$$

In this case both PEMDAS and BODMAS give the same result:
$2^3 \times 4 - 6 \times 3 ÷ (4 + 3) - 16 ÷ 4 =$
$8\times 4 - 6\times 3\div 7 - 16\div 4 = $
$32 - 18\div 7 - 16\div 4 = $(PEMDAS) or $8\times 4 - 6\times \frac 37 - 4$ (BODMAS)
$32 - \frac {18}7 - 4=$ 
$\frac {206}7 - 4 = \frac {178}7$.
The ambiguity lies when you have something like $16\div 4 \times 2$.
Whether that is supposed to be $(16\div 4)\times 2$ or $16\div (4\times 2)$.
I would say treat Multiplication and Division with equal weight and go from left to right and do $16\div 4 \times 2 = 4\times 2 = 8$, just as you'd do $16 - 4 + 2 = 12 +2 = 14$ and you'd NEVER do $16 - 4 + 2 = 16 - 6 = 10$.  EVER!
Except.... many text do give multiplication when written like this heavier weight.  So SOME text will do $16\div 4\times 2 = 16\div (4\times 2) = 16\div 8 = 2$.  
But those texts are rare and it's usually easy to tell from context what to do.
