All Operations in the Order of Operations: Parentheses within parentheses

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.

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.

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) .

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 $2$.

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.

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 is another set of inner parentheses with exponents, so we'll start by simplifying the expression inside the inner parentheses according to the aforementioned order of operations,

First, we'll calculate the numerical value of the terms with exponents, while remembering that according to the definition of a root as an exponent, the root is an exponent for all purposes, simultaneously we'll perform the multiplication in the inner parentheses and continue with the subtraction operations within these parentheses:

$\big((5^2-\sqrt{16}-7\cdot2)^2+\sqrt{81}\big):2= \\ \big((25-4-14)^2+\sqrt{81}\big):2= \\ \big(7^2+\sqrt{81}\big):2= \\$We'll continue and simplify the remaining parentheses (which were actually the outer ones), remembering that exponents come before addition and subtraction, so first we'll calculate the numerical value of the terms with exponents in the parentheses and then perform the addition operation within the parentheses:

$\big(7^2+\sqrt{81}\big):2= \\ \big(49+9\big):2=\\ 58:2=\\ 29$In the final stage, we performed the division operation,

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

$\big((5^2-\sqrt{16}-7\cdot2)^2+\sqrt{81}\big):2= \\ \big(7^2+\sqrt{81}\big):2= \\ 58:2=\\ 29$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 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.

This simple example demonstrates the order of operations, which states that multiplication and division take precedence over addition and subtraction, and that operations within parentheses take precedence over all others,

Let's consider the numerator and the denominator separately (each separately) which between them performs a division operation, meaning- we can relate to the numerator and the denominator separately as fractions in their own right, thus we can write the given fraction and write it in the following form:

$\big((3-2+4)^2-2^2\big):\frac{\sqrt{9}\cdot7}{3}= \\ \downarrow\\ \big((25-2-16)^2+3\big):\big((\sqrt{9}\cdot7):\sqrt{3} \big)$We emphasize this by noting that the fractions in the numerator and the denominator should be treated separately, indeed as if they are in their own parentheses,

Let's consider additionally that the division operation between the parentheses implies that we are dividing by the value of the denominator (meaning the denominator as a whole, it is the result of the division between the numerator and the denominator) and therefore in the given fraction to form a division that we marked for attention, the denominator being in parentheses is additionally important,

Returning to the original fraction problem, meaning - in the given form, and proceed simply,

We will start and simplify the fraction in the numerator (meaning- the numerator fraction that we are dividing by), this is done in accordance with the order of operations mentioned above, therefore we will start by calculating the numerical values of the fraction that takes precedence (this within the context of setting the root as a priority, the root being strong for everything) and then proceed with the multiplication which is in the numerator, in contrast let's consider within the parentheses that are left, those parts in the denominator are divided by the whole, they are fractions in the stronger parentheses, therefore we will also simplify this fraction, this in accordance with the order of operations mentioned above:

$\big((3-2+4)^2-2^2\big):\frac{\sqrt{9}\cdot7}{3}= \\ \big(5^2-2^2\big):\frac{3\cdot7}{3}= \\ \big(5^2-2^2\big):\frac{21}{3}\\$We will continue and simplify the fraction we received in the previous step, continue simply the fraction found within the parentheses that are divided by the whole, they are the parentheses that are left, remembering that multiplication takes precedence over addition and subtraction, therefore we will start by calculating their numerical values that take precedence in those parentheses and then proceed with the subtraction operation, in the next step the division operation of the whole (and not the division operation in the whole) takes place, and in the last step the remaining division operation takes place:

$\big(5^2-2^2\big):\frac{21}{3}=\\ \big(25-4\big):\frac{21}{3}=\\ 21:\frac{21}{3}=\\ 21:\frac{\not{21}}{\not{3}}=\\ 21:7=\\ 3$Let's consider that we advanced the division operation of the whole over the division operation in the whole itself, and this means that the number 21 in the fraction we discussed is divided by its numerical values of the whole (in its entirety)- which is the result of the division of the numerator by the denominator, therefore it was necessary to complete first the calculation of the numerical values of the whole and only then to divide the number 21 in this value,

We will conclude thus with the steps of simplifying the given fraction:
$\big((3-2+4)^2-2^2\big):\frac{\sqrt{9}\cdot7}{3}= \\ \big(5^2-2^2\big):\frac{21}{3}=\\ 21:7=\\ 3$Therefore, the correct answer is answer d'.