Solve Complex Expression: (5³:√25·3:(3·5)):(8·3-19) Step-by-Step

Let's simplify this expression while following the order of operations which states that exponents come before multiplication and division, which come before addition and subtraction, and that parentheses come before all of these,

Therefore, we'll start by simplifying the expressions in parentheses first. Note that in the expression there are two pairs of main parentheses with a division operation between them. Also note that within the left parentheses there is another pair of inner parentheses where multiplication takes place. We emphasize that since this multiplication is in (inner) parentheses, it is an operation that comes before the multiplication and division operations to its left,

We should also note that among the terms to the left of this multiplication in parentheses, there are terms with exponents (including the root which by definition of root is an exponent for all purposes). Generally, there is no prevention from calculating the numerical value of terms before/simultaneously with calculating the multiplication in the inner parentheses containing said multiplication, but for good order we will first simplify the multiplication in the inner parentheses and then calculate their numerical value, in parallel we will simplify the expression in the right parentheses, we will detail next:

$$$\big(5^3:\sqrt{25}\cdot3:(3\cdot5)\big):(8\cdot3-19)= \\ \big(5^3:\sqrt{25}\cdot3:15\big):(8\cdot3-19)= \\ \big(125:5\cdot3:15\big):(24-19)= \\ \big(125:5\cdot3:15\big):5=\\$As described above, we started by calculating the multiplication in the inner parentheses located in the left parentheses, then we calculated the numerical value of the exponential terms in those (left) parentheses, in parallel we simplified the expression in the right parentheses, this according to the aforementioned order of operations, therefore we first calculated the result of the multiplication in parentheses and only in the next stage did we calculate the result of the subtraction operation in these parentheses,

Let's continue and simplify the expression we got in the last stage, first we'll finish simplifying the expression remaining in the left parentheses, we'll do this while noting that according to the order of operations there is no precedence between multiplication and division operations in this expression therefore we will calculate this expression's result step by step from left to right, which is the natural order of operations, in the final stage we will divide the expression obtained from simplifying the expression in the left parentheses by the term we got from simplifying the right parentheses in the previous stage:

$\big(125:5\cdot3:15\big):5=\\ \big(25\cdot3:15\big):5=\\ \big(75:15\big):5=\\ 5:5=\\ 1$Let's summarize then the stages of simplifying the given expression, we got that:

$\big(5^3:\sqrt{25}\cdot3:(3\cdot5)\big):(8\cdot3-19)= \\ \big(125:5\cdot3:15\big):5=\\ \big(75:15\big):5=\\ 5:5=\\ 1$Therefore the correct answer is answer D.

Note:

Note that the parentheses we called throughout the above solution "the left parentheses" are actually redundant, this is because between the terms within them (including the expression in the inner parentheses, which we currently refer to in this note as one term, meaning - the result of this multiplication), there is no operation that comes after the division operation applying to these parentheses (meaning - according to the aforementioned order of operations) and as seen in the above solution, the operations (both within these parentheses and when applying the division operation on them) were performed anyway in left to right order therefore we could have omitted these (left) parentheses from the start and gotten the same result, in fact the calculation was identical in all its stages, except for the presence of these (meaningless) parentheses in the expression, however in the original problem parentheses define order of operations and therefore we'll act according to them.