Evaluate [7² - (5+4)] ÷ [(3² - 2³)¹⁴ + 7] × 3: Complete Solution Guide

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, note that in this expression there are two pairs of parentheses with a division operation between them and a number multiplying them, additionally note that within each of these pairs of parentheses mentioned above there is another pair of parentheses and one of them has an exponent, so we'll start by simplifying each of the expressions within the inner parentheses:

$\big(7^2-(5+4)\big):\big((3^2-2^3)^{14}+7\big)\cdot3= \\ \big(7^2-9\big):\big((9-8)^{14}+7\big)\cdot3= \\ \big(7^2-9\big):\big(1^{14}+7\big)\cdot3\\$We simplified the expressions within the inner parentheses that are in the two pairs of outer parentheses (between which the division operation is performed), we did this by performing the addition operation in the left (inner) parentheses, in the right (inner) parentheses since exponents come before addition and subtraction, we first calculated the numerical value of the terms with exponents and then performed the subtraction operations,

We'll continue and simplify the expression obtained in the last step by simplifying the expressions in parentheses, again we'll prioritize calculating the numerical value of terms with exponents before their addition and subtraction operations:

$\big(7^2-9\big):\big(1^{14}+7\big)\cdot3=\\ \big(49-9\big):\big(1+7\big)\cdot3=\\ 40:8\cdot3$Now note that between multiplication and division operations there is no defined order of operations, meaning- neither operation takes precedence, also in the expression obtained in the last step there are no parentheses dictating a specific order, therefore we'll calculate the value of the expression obtained at this stage, step by step from left to right, which is the natural order of operations in the absence of operation precedence, first we'll perform the division operation and then the multiplication operation:

$40:8\cdot3 =\\ 5\cdot3 =\\ 15$Let's summarize the steps of simplifying the given expression, we got that:

$\big(7^2-(5+4)\big):\big((3^2-2^3)^{14}+7\big)\cdot3= \\ \big(7^2-9\big):\big(1^{14}+7\big)\cdot3\\ 40:8\cdot3 =\\ 15$Therefore the correct answer is answer A.