Complex Expression Comparison: Evaluating 1/9((5²-8÷2)÷7)² vs. (-49)(-2)+3÷3

According to the given problem, whether it is discussed in addition or in subtraction each of the terms that comes in its turn,

this is done within the framework of the order of operations, which states that multiplication precedes addition and subtraction, and that the preceding operations are performed before all others,

A. We will start with the terms that are on the left in the given problem:

$\frac{1}{9}\cdot\big((5^2-8:2):7\big)^2$Begin by simplifying the terms that are in parentheses (the denominators), which are prioritized in strength, the multiplication divides the break in accordance with the order of operations mentioned, note that the terms in the denominators include within them the operation of division part of the term in parentheses (the numerators), therefore we will start simply with this term, in this term the operation of subtraction is performed between a numerator of greater strength to a numerator of lesser strength, therefore the beginning will execute the calculation of their numerical values which in the course of being executed the operation of division and continues to perform the operation of subtraction, in the last stage the operation of division part on the term in parentheses is executed:

$\frac{1}{9}\cdot\big((5^2-8:2):7\big)^2= \\ \frac{1}{9}\cdot\big((25-8:2):7\big)^2 = \\ \frac{1}{9}\cdot\big((25-4):7\big)^2= \\ \frac{1}{9}\cdot\big(21:7\big)^2=\\ \frac{1}{9}\cdot3^2=\\$note that there is no prohibition to calculate their numerical values of the numerator that is stronger which in the term in parentheses in comparison to calculate the numerators which in the term in parentheses, this means that breaking in separate numerators, even for the sake of good order we performed this stage after stage,

We will continue simply the term we received in the last stage, we remember that multiplication precedes division and therefore we will start from calculating their numerical values of the numerator that is stronger, in the next stage the multiplication in the break is executed, and at the end the operation of division (by summarizing the break received):

$\frac{1}{9}\cdot3^2=\\ \frac{1}{9}\cdot9=\\ \frac{1\cdot9}{9}=\\ \frac{\not{9}}{\not{9}}=\\ 1$In the last stages we performed the multiplication of the number 9 in the break, this we performed within that we remember that the multiplication in the break means the multiplication in the amount of the break,

We finished simply the term that is on the left in the given problem, we will summarize the stages of the simplification:

We received that:

$\frac{1}{9}\cdot\big((5^2-8:2):7\big)^2= \\ \frac{1}{9}\cdot\big((25-4):7\big)^2= \\ \frac{1}{9}\cdot3^2=\\ 1$

B. We will continue and simplify the term that is on the right in the given problem:

$\big((-49)\cdot(-2)+3:3\big)\cdot\frac{1}{99}$In this part to perform in the first part simplify the term within the framework on the order of operations,

In this term the operation of division part on the term in parentheses is established, therefore we will simplify starting with this term, we remember that multiply and division precede addition, therefore we will calculate starting their numerical values of the multiplication in the term (this within that we remember that multiplying a series of numbers in a series of numbers gives a mandatory result) and in comparison we will calculate the result of the operation of division in the term, in the next stage the operation of addition in the term is executed:

$\big((-49)\cdot(-2)+3:3\big)\cdot\frac{1}{99} \\ \big(98+1\big)\cdot\frac{1}{99} =\\ 99\cdot\frac{1}{99} \\$We will continue and execute the multiplication in the break, this within that we remember that the multiplication in the break means the multiplication in the amount of the break, in the next stage the operation of division of the break is executed (by summarizing the break):

$99\cdot\frac{1}{99} \\ \frac{ 99\cdot1}{99} \\ \frac{\not{99}}{\not{99}}=\\ 1$We finished simply the term that is on the right in the given problem, we will summarize the stages of the simplification:

We received that:

$\big((-49)\cdot(-2)+3:3\big)\cdot\frac{1}{99} \\ 99\cdot\frac{1}{99} =\\ 1$We will return now to the original problem, and we will present the results of the simplification of the terms that were reported in A and in B:

$\frac{1}{9}\cdot\big((5^2-8:2):7\big)^2\textcolor{red}{☐}\big((-49)\cdot(-2)+3:3\big)\cdot\frac{1}{99} \\ \downarrow\\ 1\textcolor{red}{☐}1$As a result that is established that:

$1 \text{ }\textcolor{red}{=}1$Therefore the correct answer here is answer B.