Solve [(136-√144)÷2³×2]÷15: Complete Order of Operations Challenge

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, noting that in the expression there are parentheses with division operations, and also note that within these parentheses there are inner parentheses that also have division operations, so we'll start by simplifying the expression inside the inner parentheses according to the order of operations mentioned above,

$\big((136-\sqrt{144}):2^3\cdot2\big):(3\cdot5)= \\ \big((136-12):2^3\cdot2\big):(3\cdot5)= \\ \big(124:2^3\cdot2\big):(3\cdot5)=\\$In the first stage, we calculated the numerical value of the root in the inner parentheses, and then we performed the subtraction operation within these parentheses,

For good order, we will simplify the expression in the left parentheses first and only then simplify the expression in the right parentheses,

We'll continue then and simplify the expression in the (left) parentheses that remained in the expression we got in the last stage, note that in this expression there are division operations and exponents, so according to the order of operations we'll first calculate the numerical value of the exponent and then perform the division and multiplication operations step by step from left to right:

$\big(124:2^3\cdot2\big):(3\cdot5)=\\ \big(124:8\cdot2\big):(3\cdot5)=\\ \big(\frac{124}{8}\cdot2\big):(3\cdot5)=$In the final stage, since we have division and multiplication operations where the order of operations does not $$define precedence for either and also there are no parentheses defining such precedence, we started calculating the expression result according to the natural order of operations, meaning - from left to right, additionally, since the division operation doesn't yield a whole number result, we converted this division to a fraction (an improper fraction in this case, since the numerator is larger than the denominator), then we'll perform this division operation by reducing the fraction and at stage:

$\big(\frac{\not{124}}{\not{8}}\cdot2\big):(3\cdot5)=\\ \big(\frac{31}{2}\cdot2\big):(3\cdot5)=\\ \big(\frac{31\cdot2}{2}\big):(3\cdot5)\\$In the final stage, after reducing the fraction, we performed the multiplication by the fraction, while remembering that multiplication by a fraction means multiplication by the fraction's numerator, then - we'll reduce the new fraction that resulted from the multiplication operation again, and in the stage after that we'll perform the multiplication in the right parentheses - which we haven't dealt with yet:

$\big(\frac{31\cdot\not{2}}{\not{2}}\big):(3\cdot5)=\\ 31:15\\$ Note that the reduction operation (which is essentially a division operation) could only be performed because there is multiplication between the terms in the numerator,

We'll now finish simplifying the given expression, meaning - we'll perform the remaining division operation, again, since this division operation doesn't yield a whole number result, we'll first convert this division to an improper fraction and then convert it to a mixed number, by taking out the whole numbers (meaning the number of complete times the denominator goes into the numerator) and adding the remainder divided by the divisor (15):

$31:15=\\ \frac{31}{15}=\\ 2\frac{1}{15}$In the final stages we performed the multiplication operation in the right parentheses and finally performed the division operation, note that there was no prevention from calculating the multiplication result in the right parentheses from the first stage, which we carried through the entire simplification until this stage, however as mentioned before, for good order we preferred to do this in the final stage,

Let's summarize the stages of simplifying the given expression:

$\big((136-\sqrt{144}):2^3\cdot2\big):(3\cdot5)= \\ \big(124:2^3\cdot2\big):(3\cdot5)=\\ \big(\frac{124}{8}\cdot2\big):(3\cdot5)=\\ \big(\frac{31}{2}\cdot2\big):(3\cdot5)=\\ \frac{31}{15}=\\ 2\frac{1}{15}$Therefore the correct answer is answer B.