Parentheses in advanced Order of Operations: Parentheses within parentheses

In the order of operations, parentheses come before everything else.

We start by solving the inner parentheses in the subtraction operation:

$((3):3-1)\times4=$ We continue with the inner parentheses in the division operation and then subtraction:

$(1-1)\times4=$

We continue solving the subtraction exercise within parentheses and then multiply:

$0\times4=0$

We begin by simplifying the given expression paying attention to the order of arithmetic operations which states that powers precede multiplication, division precedes addition and subtraction and that parentheses precede all of the above.

Let's keep in mind that in the given expression there are no parentheses or powers. However there are multiplication and division operations, so we will use them as our starting point. After which we will perform the addition and subtraction operations:

$27:3-9\cdot2+5+3= \\ 9-18+5+3=\\ -1$Therefore, the correct answer is option B.

This simple rule is the order of operations which states that multiplication and division come before addition and subtraction, and operations enclosed in parentheses come first,

In the given example of division between two given numbers in parentheses, therefore according to the order of operations mentioned above, we start by calculating the values of each of the numbers within the parentheses, there is no prohibition against calculating the result of the addition operation in the given number, for the sake of proper order, this operation is performed later:

$(3+2-1):(1+3)-1+5= \\ 4:4-1+5$In continuation of the principle that division comes before addition and subtraction the division operation is performed first and then the operations of subtraction and addition which were received in the given number and in the last stage:

$4:4-1+5= \\ 1-1+5=\\ 5$Therefore the correct answer here is answer B.

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. In this problem, there are parentheses within parentheses, so we'll first deal with the inner parentheses, simplify the expression within them and continue similarly with the outer parentheses and simplify the expression within them. We'll do all of the above while maintaining the order of operations mentioned at the beginning of the solution:

$3+[5\cdot4-(9\cdot1+3)]-8= \\ 3+[5\cdot4-(9+3)]-8= \\ 3+[5\cdot4-12]-8= \\ 3+[20-12]-8= \\ 3+8-8= \\ 3$

Note that since multiplication and division come before addition and subtraction, in the first stage, we first performed the multiplication in the inner parentheses and then performed the addition, and continued similarly with the expression we got after opening these parentheses which was in the outer parentheses, first we performed the multiplication in the first term in the outer parentheses, and then the subtraction within them, finally, after opening the outer parentheses we performed the addition and subtraction operations,

Therefore we got that answer C is the correct answer.

This expression is simplified while maintaining the order of operations which states that parentheses take come before exponents, and the exponents come before multiplication and division which come before addition and subtraction.

Therefore, we will start first by simplifying the expressions in the parentheses, in this case there are parentheses within parentheses, so we will first deal with the inner parentheses.

We will simplify the expression within the innermost parentheses and then we will perform the exponentiation on them, then we will deal similarly with the outer parentheses while maintaining the order of operations:

$\big((5-4\cdot3)^2+8-3\big):2= \\ \big((5-12)^2+8-3\big):2= \\ \big((-7)^2+8-3\big):2= \\ \big(49+8-3\big):2=\\$Note that since exponents come before multiplication and division we first performed the exponentiation in the outer parentheses and then the division operation, we continued and performed first the exponentiation of the expression results in the inner parentheses by squaring. (Remember that raising any number (positive or negative) to an even, positive power will always give a positive result).

We continue and finish dealing with the expression in the remaining parentheses, then we perform the division operation that applies to the parentheses:

$\big(49+8-3\big):2=\\ 54:2=\\ 27$In summary of the solution steps, we found that:

$\big((5-4\cdot3)^2+8-3\big):2= \\ \big((5-12)^2+8-3\big):2= \\ \big(49+8-3\big):2=\\ 27$Therefore, the correct answer is answer a.

To solve a problem given in division or multiplication each of the terms that appear in its expression separately,

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 appear on the left in the given problem:

$\frac{1}{9}\cdot((4^2-3\cdot2):2+4)$First, we simplify the terms in the parentheses (the numerators) that multiply the fraction according to the order of operations, noting that the term in the parentheses includes within it an operation of division of the term in the parentheses (the denominators), therefore, we will start simply with this term, in this term a subtraction operation is performed between terms that strengthens the division between terms, therefore the calculation of its numerical value is carried out first followed by the multiplication of the terms and continue to perform the subtraction operation:

$\frac{1}{9}\cdot((4^2-3\cdot2):2+4) =\\ \frac{1}{9}\cdot((16-3\cdot2):2+4) =\\ \frac{1}{9}\cdot((16-6):2+4) =\\ \frac{1}{9}\cdot(10:2+4)\\$Note that there is no prohibition to calculate their numerical values of the term that strengthens as in the term in the parentheses in contrast to the multiplication that in the term in the parentheses, this from a concept that breaks in separate terms, also for the sake of good order we performed this step after step,

We continue simply with the terms in the parentheses that were left, we remember that division precedes subtraction and therefore we will start from calculating the outcome of the multiplication in the term, in the next step the division is performed and finally the multiplication in the break that multiplies the term in the parentheses:

$\frac{1}{9}\cdot(10:2+4)\\ \frac{1}{9}\cdot(5+4)=\\ \frac{1}{9}\cdot9=\\ \frac{1\cdot9}{9}=\\ \frac{\not{9}}{\not{9}}=\\ 1$In the last steps we performed the multiplication of the number 9 in the break, this we did while we remember that the multiplication in the break means the multiplication in the amount of the break,

We finished simply with the terms that appear on the left in the given problem, we will summarize the steps of the simplification:

We received that:

$\frac{1}{9}\cdot((4^2-3\cdot2):2+4) =\\ \frac{1}{9}\cdot(10:2+4)\\ \frac{1}{9}\cdot(5+4)=\\ \frac{9}{9}=\\ 1$

B. We will continue and simplify the terms that appear on the right in the given problem:

$4^2-(3\cdot2:2+4)\cdot\frac{1}{7}$In this part to be done in the first part we simplify the terms within the framework of the order of operations,

In this term a multiplication operation is performed on the term in the parentheses, therefore, we will simplify first this term, we remember that multiplication and division precede subtraction, therefore, we will calculate first the numerical values of the first term from the left in this term, noting that the concept that between multiplication and division there is no predetermined precedence in the order of operations, the operations in this term are performed one after the other according to the order from left to right, which is the natural order of operations, in contrast we will calculate the numerical values of the term that strengthens:

$4^2-(3\cdot2:2+4)\cdot\frac{1}{7} =\\ 16-(6:2+4)\cdot\frac{1}{7} =\\ 16-(3+4)\cdot\frac{1}{7} =\\ 16-7\cdot\frac{1}{7}\$We will continue and perform 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 step the division operation of the break (by the compression of the break) is performed and in the last step the remaining subtraction operation, this in accordance with the order of operations:

$16-7\cdot\frac{1}{7}=\\ 16-\frac{7\cdot1}{7}=\\ 16-\frac{\not{7}}{\not{7}}=\\ 16-1=\\ 15$We finished simply with the terms that appear on the right in the given problem, we will summarize the steps of the simplification:

We received that:

$4^2-(3\cdot2:2+4)\cdot\frac{1}{7} =\\ 16-(3+4)\cdot\frac{1}{7} =\\ 16-1=\\ 15$We will return to the original problem, and we will present the outcomes of the simplifications that were reported in A and B:

$\frac{1}{9}\cdot((4^2-3\cdot2):2+4)\textcolor{red}{☐}4^2-(3\cdot2:2+4)\cdot\frac{1}{7} \\ \downarrow\\ 1 \textcolor{red}{☐}15$As a result that is established that:

$1 \text{ }\textcolor{red}{\neq}15$Therefore, the correct answer here is answer B.

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, noting that in this problem there are two pairs of parentheses, so we'll simplify gradually, starting with the expression inside the left parentheses in the first term on the left and perform the addition within them, while simultaneously simplifying the expression in the right parentheses, following the order of operations mentioned above, so we'll perform the multiplication in them first:

$(124+8):4+16:2+(12\cdot3-3):11= \\ 132:4+16:2+(36-3):11$

Let's continue simplifying the expression, first completing the simplification of the remaining right parentheses by performing the subtraction operation, and simultaneously performing the division operations in the first two terms of the expression we got in the last step, leaving the addition operations between all terms for the end in accordance with the order of operations:

$132:4+16:2+(36-3):11 =\\ 33+8+33:11$

Let's continue, again performing first the division operation in the third term from the left and finally performing the addition operations:

$33+8+33:11 =\\ 33+8+3=\\ 44$

To summarize the simplification steps, we got that:

$(124+8):4+16:2+(12\cdot3-3):11= \\ 33+8+3=\\ 44$

Therefore, the correct answer is answer D.

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 them,

Therefore, we'll start by simplifying the expressions in parentheses, noting that in this expression there are parentheses within parentheses, so we'll begin by simplifying the expression in the innermost parentheses and perform the addition operation within them:

$96-64:\big((5+13):9\cdot4\big)+48= \\ 96-64:\big(18:9\cdot4\big)+48$

Next, we'll perform the division and multiplication operations that remain in the parentheses, we'll do this according to the natural order of operations from left to right, since there are no additional parentheses in this expression (within the parentheses) dictating a different order of operations, and between these two operations there is no defined order as mentioned above, therefore we'll first perform the division operation and then the multiplication operation:

$96-64:\big(18:9\cdot4\big)+48=\\ 96-64:\big(2\cdot4\big)+48=\\ 96-64:8+48$

We'll continue and remember that multiplication and division come before addition and subtraction, therefore we'll first perform the division operation and only in the next step the subtraction and addition operations:

$96-64:8+48 =\\ 96-8+48 =\\ 136$

Let's summarize the steps of simplifying the expression:

$96-64:\big((5+13):9\cdot4\big)+48= \\ 96-64:\big(18:9\cdot4\big)+48=\\ 96-64:8+48 = \\ 136$

Therefore, the correct answer is answer B.

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 parentheses within parentheses, where the first parentheses from the left are raised to a power and between the two pairs of parentheses there is a division operation, therefore, first we'll simplify the expressions in both pairs of parentheses, we'll do this according to the order of operations, first we'll calculate the values of the expressions with exponents (in both pairs of parentheses) then we'll calculate the result of multiplication in the first parentheses from the left and then we'll calculate the results of addition and subtraction operations in both pairs of parentheses:

$\big((2\cdot3^2+5)^2:(4^2+3^2-2)\big):23= \\ \big((2\cdot9+5)^2:(16+9-2)\big):23= \\ \big((18+5)^2:(16+9-2)\big):23=\\ \big(23^2:23\big):23$

Next we'll apply the exponent to the result of simplifying the expression in the first parentheses from the left, this is according to the aforementioned order of operations, and then we'll perform the mentioned division operation within the remaining large parentheses, finally - we'll perform the division operation that applies to the parentheses:

$\big(23^2:23\big):23 =\\ \big(529:23\big):23 =\\ 23:23=\\ 1$

Let's summarize the result of simplifying the expression, we got that:

$\big((2\cdot3^2+5)^2:(4^2+3^2-2)\big):23= \\ \big((2\cdot9+5)^2:(16+9-2)\big):23= \\ \big(23^2:23\big):23 =\\ 1$

Therefore the correct answer is answer A.

Note:

The final steps can of course be calculated numerically, step by step as described there, but note that we can also reach the same result without calculating the numerical value of the terms in the expression, by using the law of exponents for terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$

We'll do this in the following way:
$\big(23^2:23\big):23 =\\ \frac{23^2}{23}:23=\\ 23^{2-1}:23=\\ 23:23=\\ 1$

First we converted the division operation in parentheses to a fraction, then we applied the aforementioned law of exponents while remembering that any number can be represented as itself to the power of 1 (and that any number to the power of 1 equals the number itself) and finally we remembered that dividing any number by itself will always give the result 1.

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, additionally note that inside the left parentheses there is another pair of parentheses with an exponent, so we'll start by simplifying the expression within the inner parentheses that are inside the left parentheses:

$\big(\big((-2)^3+2^4\big)^2:4+2^3\cdot3\big):(4\cdot5)= \\ \big(\big(-8+16\big)^2:4+2^3\cdot3\big):(4\cdot5)= \\ \big(8^2:4+2^3\cdot3\big):(4\cdot5)\\$We simplified the expression in the inner parentheses on the left, this was done in two steps because there was an addition operation between two terms with exponents, therefore, according to the order of operations mentioned above, we first calculated the numerical values of the terms with exponents, this was done while remembering that raising an odd number to a power maintains the sign of the number being raised, then we performed the addition operation within the (inner) parentheses,

Let's continue, for good order, we'll simplify the expression in the left parentheses first and only then simplify the expression in the right parentheses, let's remember again the order of operations mentioned above, therefore we'll start by calculating the terms with exponents since exponents come before multiplication and division, then we'll perform the division and multiplication operations within these parentheses and finally we'll perform the addition operation within the parentheses:

$\big(8^2:4+2^3\cdot3\big):(4\cdot5)=\\ \big(64:4+8\cdot3\big):(4\cdot5)=\\ \big(64:4+8\cdot3\big):(4\cdot5)=\\ \big(16+24\big):(4\cdot5)=\\ 40:20=\\ 2$In the final stages we performed the multiplication within the right parentheses and finally performed the division operation, note that there was no prevention from the first stage to calculate the result of the multiplication in the right parentheses, 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:

$\big(\big((-2)^3+2^4\big)^2:4+2^3\cdot3\big):(4\cdot5)= \\ \big(8^2:4+2^3\cdot3\big):(4\cdot5)=\\ \big(64:4+8\cdot3\big):(4\cdot5)=\\ 40:20=\\ 2$Therefore the correct answer is answer C.

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.

This simple example demonstrates the order of operations, which states that exponentiation precedes multiplication and division, which come before addition and subtraction, and that operations within parentheses come first,

In the given example, the operation of division between parentheses (the denominators) by a number (which is also in parentheses but only for clarification purposes), thus according to the order of operations mentioned we start with the parentheses that contain the denominators first, this parentheses that contain the denominators includes multiplication between two numbers which are also in parentheses, therefore according to the order of operations mentioned, we start with the numbers inside them, paying attention that each of these numbers, including the ones in strength, and therefore assuming that exponentiation precedes multiplication and division we consider their numerical values only in the first step and only then do we perform the operations of multiplication and division on these numbers:

$$$$$\lbrack(3^2-4-5)\cdot(4+\sqrt{16})-5 \rbrack:(-5)=\\ \lbrack(9-4-5)\cdot(4+4)-5 \rbrack:(-5)=\\ \lbrack0\cdot8-5 \rbrack:(-5)\\$Continuing with the simple division in parentheses ,and according to the order of operations mentioned, we proceed from the multiplication calculation and remember that the multiplication of the number 0 by any number will yield the result 0, in the first step the operation of subtraction is performed and finally the operation of division is initiated on the number in parentheses:

$\lbrack0\cdot8-5 \rbrack:(-5)= \\ \lbrack0-5 \rbrack:(-5)= \\ -5 :(-5)=\\ 1$ Therefore, the correct answer is answer c.

We will simplify this expression while maintaining the order of operations which states that parentheses come before multiplication and division,which come before addition and subtraction.

Let's start first by simplifying the expressions in the parentheses, we will note that in this expression there are two pairs of parentheses between which multiplication takes place.

Notice that the left inner parentheses are raised to a power, so let's start simplifying the expression which is within the inner parentheses.

$\big((8^2-3+5^2+7\cdot2)^2:100 \big)\cdot(100:10)=\\ \big((64-3+25+14)^2:100 \big)\cdot(100:10)=\\ \big(100^2:100 \big)\cdot(100:10)$We simplified the expression which is in the inner parentheses found within the left parentheses.

We did this in two steps because there are addition and subtraction operations between terms in parentheses, and there is also multiplication of terms (according to the order of operations, we first calculated the terms in parentheses, then we calculated the result of the multiplication in these parentheses and then we performed the addition and subtraction operations which are in the parentheses).

Then, we will simplify the expression which is in the left parentheses first, and only then we will simplify the expression which is in the right parentheses.

We will start by calculating the term in parentheses since parentheses precede multiplication and division, then we will perform the division operation which is in the parentheses:

$\big(100^2:100 \big)\cdot(100:10)=\\ \big(10000:100 \big)\cdot(100:10)=\\ 100\cdot(100:10)=\\ 100\cdot10=\\ 1000$In the last steps we divided within the right set of parentheses and finally we multiplied.

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

$\big((8^2-3+5^2+7\cdot2)^2:100 \big)\cdot(100:10)=\\ \big(100^2:100 \big)\cdot(100:10)=\\ 100\cdot(100:10)=\\ 100\cdot10=\\ 1000$Therefore the correct answer is answer C.

Note:

The expression in the left parentheses in the last steps can be calculated numerically step by step as described there, but note that it is also possible to reach the same result without calculating their numerical value of the terms in the expression, by using the law of exponents to give terms with identical bases:

$\frac{a^m}{a^n}=a^{m-n}$ This is done as follows:
$100^2:100=\\ \frac{100^2}{100}=\\ 100^{2-1}=\\ 100$First we converted the division operation to a fraction, then we applied the above law of exponents while remembering that any number can be represented as the same number to the power of 1 (and any number to the power of 1 equals the number itself) .

To solve the expression $[(4-2)^2]^3$, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

Step 1: Solve the innermost parentheses:

The expression inside the innermost parentheses is $4-2$. We perform the subtraction:

$4-2 = 2$

Step 2: Apply the exponentiation:

Next, we take the result of the subtraction and apply the squaring operation $((2)^2)$:

$2^2 = 4$

Step 3: Apply the outer exponentiation:

Finally, we take the result of the previous step and raise it to the power of 3:

$4^3 = 64$

Therefore, the value of the expression $[(4-2)^2]^3$ is $64$.

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 them,

Therefore, we'll start by simplifying the expressions in parentheses, we'll do this according to the order of operations mentioned above, first we'll calculate the numerical value of the term with the exponent, which is the divided term in parentheses, and simultaneously we'll calculate the result of the multiplication in parentheses, then we'll calculate the addition in them:

$65-(4\cdot3+2^3:4)-4^3+5^2:5= \\ 65-(12+8:4)-4^3+5^2:5=\\ 65-(12+2)-4^3+5^2:5=\\ 65-14-4^3+5^2:5=\\$We'll continue and simplify the resulting expression, first we'll calculate the numerical value of the term with the exponent, which is the third term from the left, simultaneously we'll calculate the numerical value of the term with the exponent which is divided by the fourth term from the left and we'll continue to perform the division operation on this term, this is because multiplication and division come before addition and subtraction:

$65-14-4^3+5^2:5=\\ 65-14-64+25:5=\\ 65-14-64+5$$$We'll finish simplifying the given expression and perform the addition and subtraction operations:

$65-14-64+5 =\\ -8$Let's summarize the steps of simplifying the given expression, we got that:

$65-(4\cdot3+2^3:4)-4^3+5^2:5= \\ 65-(12+8:4)-4^3+5^2:5=\\ 65-14-64+5 =\\ -8$Therefore, the correct answer is answer C.

Simplifying this expression emphasizes the order of operations, which states that multiplication precedes addition and subtraction, and that division precedes all of them,

In the given expression, the establishment of division operations between the parentheses (the outermost) to a number, therefore according to the order of operations as mentioned, is handled by simplifying the expression in these parentheses, this expression includes division operations that begin on the expression within the parentheses (the innermost), therefore according to the order of operations as mentioned is handled by simplifying the expression in these parentheses and performing the subtraction operations in it, there is no hindrance to calculate the outcome of the division operations in the expression in the outermost parentheses, but for the sake of good order this is done afterwards:

$\lbrack(4+3):7+2:2-2\rbrack:5= \\ \lbrack7:7+2:2-2\rbrack:5$Continuing and simplifying the expression in the parentheses we noted, since division precedes addition and subtraction, start with the division operations in the expression and only then calculate the outcome of the addition and subtraction, ultimately perform the division operations on this expression in the parentheses:

$\lbrack7:7+2:2-2\rbrack:5 \\ \lbrack1+1-2\rbrack:5=\\ 0:5=\\0$In the last stage we mentioned that multiplying a number by 0 gives the result 0,

Therefore, this simplifying expression is short so there is no need to elaborate,

And the correct answer here is answer A.

According to the rules of order of arithmetic operations, parentheses are resolved first.

We start by solving the inner parentheses, first we will solve the root using the formula:

$\sqrt{a}=\sqrt{a^2}=a$

$\sqrt{81}=\sqrt{9^2}=9$

The exercise obtained within parentheses is:

$(9-3\times3)$

First we solve the multiplication exercise and then we subtract:

$(9-9)=0$

After solving the inner parentheses, the resulting exercise is:

$0:4+5\times5$

According to the rules of the order of arithmetic operations, we first solve the exercises of multiplication and division, and then subtraction.

We place the two exercises within parentheses to avoid confusion:

$(0:4)+(5\times5)=0+25=25$

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 two pairs of parentheses with subtraction between them, and also that within the left parentheses there are two more pairs of inner parentheses with division between them, so we'll start by simplifying the expressions within the parentheses, both the expression in the inner parentheses with division between them that are inside the outer left parentheses, and the expression in the right parentheses, this is done according to the order of operations mentioned above:

$\big((3^4-8+5):(6^2+\sqrt{9})\big)-(\sqrt{64}-4^2)= \\ \big((81-8+5):(36+3)\big)-(8-16)= \\ 78:39-(-8)\\$We simplified the above expressions (those within the parentheses) while remembering that exponents come before addition and subtraction, so first we calculated the numerical values of the terms with exponents (while remembering that according to the definition of root as an exponent, the root is an exponent for all purposes) and then we performed the addition and subtraction operations within the parentheses,

In the final stage, since the result of the subtraction operation in the right parentheses yielded a negative result, we kept this result in parentheses, which we will open in the next stage, while remembering that according to the multiplication law, multiplying a negative number by a negative number gives a positive result,

Let's continue then and simplify the expression we got in the last stage:

$78:39-(-8)=\\ 2+8=\\ 10$We simplified the expression where in the first stage we performed the division operation in the first term from the left and simultaneously opened the parentheses on the right (this is according to the multiplication law as mentioned before) and this is because multiplication and division come before addition and subtraction, then we performed the addition operation,

Therefore the correct answer is answer A.

This simple rule is the emphasis on the order of operations which states that exponentiation precedes multiplication and division, which precede addition and subtraction, and that operations within parentheses precede all others,

Let's consider that the numerator is the whole and the denominator is the part which breaks (every break) into whole pieces (in their entirety) among which division operation is performed, meaning- we can relate the numerator and the denominator of the break as whole pieces in closures, thus we can express the given fraction and write it in the following form:

$\frac{(25-2-16)^2+3}{8+5}:\sqrt{9}= \\ \downarrow\\ \big((25-2-16)^2+3\big):(8+5):\sqrt{9}$We highlight this by noting that fractions in the numerator of the break and in its denominator are considered separately, as if they are in closures,

Let's return to the original fraction in question, meaning - in the given form, and simplify, separately, the fraction in the numerator of the break which causes it and the fraction in its denominator, this is done in accordance to the order of operations mentioned and in a systematic way,

Let's consider that in the numerator of the break the fraction we get changes into a fraction in closures which indicates strength, therefore we will start simplifying this fraction, given that this fraction includes only addition and subtraction operations, perform the operations in accordance to the natural order of operations, meaning- from left to right, simplifying the fraction in the numerator of the break:

$\frac{(25-2-16)^2+3}{8+5}:\sqrt{9}=\\ \frac{7^2+3}{13}:\sqrt{9}\\$We will continue and simplify the fraction we received in the previous step, this of course, in accordance to the natural order of operations (which states that exponentiation precedes multiplication and division, which precede addition and subtraction, and that operations within parentheses precede all others), therefore we will start from calculating the numerical values of the exponents in strength (while we remember that in defining the root as strength, the root itself is strength for everything), and then perform the division operation which is in the numerator of the break:

$\frac{7^2+3}{13}:\sqrt{9}=\\ \frac{49+3}{13}:3=\\ \frac{52}{13}:3\\$We will continue and simplify the fraction we received in the previous step, starting with performing the division operation of the break, this is done by approximation, and then perform the remaining division operation:

$\frac{\not{52}}{\not{13}}:3=\\ 4:3=\\ \frac{4}{3}$In the previous step, given that the outcome of the division operation is different from a whole (greater than whole for the numerator, given that the divisor is greater than the dividend) we marked its outcome as a fraction in approximation (where the numerator is greater than the denominator),

We conclude the steps of simplifying the given fraction, we found that:

$\frac{(25-2-16)^2+3}{8+5}:\sqrt{9}=\\ \frac{7^2+3}{13}:\sqrt{9}=\\ \frac{52}{13}:3=\\ \frac{4}{3}$Therefore, the correct answer is answer b'.

Note:

Let's consider that in the group of the previous steps in solving the problem, we can start recording the break and the division operation that affects it even without the break, but with the help of the division operation:

$\frac{52}{13}:3\\ \downarrow\\ 52:13:3$And from here on we will start calculating the division operation in the break and only after that we performed the division in number 3, we emphasize that in general we simplify this fraction in accordance to the natural order of operations, meaning we perform the operations one after the other from left to right, and this means that there is no precedence of one division operation in the given fraction over the other except as defined by the natural order of operations, meaning- in calculating from left to right, (Let's consider additionally that defining the order of operations mentioned at the beginning of the solution, which states that exponentiation precedes multiplication and division, which precede addition and subtraction, and that operations within parentheses precede all others, does not define precedence even among multiplication and division, and therefore the judgment between these two operations, in different closures, is in a different order, it is in calculating from left to right).

To solve the problem $[(136-\sqrt{144}):2^3\cdot2-1]:(3\cdot5) =$, we need to apply the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Start by calculating the expression inside the innermost parentheses:

• First, evaluate the square root: $\sqrt{144} = 12$. Substitute back into the expression, giving us $(136 - 12)$.

• Subtract: $136 - 12 = 124$.

• The expression now becomes $[124:2^3\cdot2-1]:(3\cdot5)$.

Next, handle the exponents:

• Calculate $2^3 = 8$.

Substitute back, and the expression becomes:

• $[124:8\cdot2-1]:(3\cdot5)$

Now perform the operations inside the square brackets:

• Perform the division: $124:8 = 15.5$.

• Next, multiply: $15.5 \cdot 2 = 31$.

• Subtract 1: $31 - 1 = 30$.

The expression simplifies to:

• $30:(3\cdot5)$

Simplify further by handling the multiplication in the denominator:

• Calculate $3 \cdot 5 = 15$.

Finally, divide:

• $30:15 = 2$.

Thus, the answer is $2$.