In previous articles, we have seen what is the order of operations for addition, subtraction, multiplication, and division and also the order we must follow when there are exponents.

When the exercise we need to solve includes parentheses, we always (always!) start with the operation contained within them.

Reminder: when an exercise presents operations that have the same precedence, that is, multiplications and divisions or additions and subtractions, we will solve the exercise from left to right.

## Examples with solutions for Parentheses in advanced Order of Operations

### Exercise #1

$8\times(5\times1)=$

### Step-by-Step Solution

According to the order of operations, we first solve the expression in parentheses:

$5\times1=5$

Now we multiply:

$8\times5=40$

40

### Exercise #2

$(7+2)\times(3+8)=$

### Step-by-Step Solution

Simplify this expression paying attention to the order of operations. Whereby exponentiation precedes multiplication, division precedes addition and subtraction and that parentheses precede all of the above.

Therefore, let's first start by simplifying the expressions within the parentheses. After which we perform the multiplication between them:

$(7+2)\cdot(3+8)= \\ 9\cdot11=\\ 99$Therefore, the correct answer is option B.

99

### Exercise #3

$(12-6+9)\times(7+3)=$

### Step-by-Step Solution

According to the order of operations, we will first solve the expressions in parentheses, and then multiply:

$(12-6+9)=(6+9)=15$

$(7+3)=10$

Now let's solve the multiplication problem:

$15\times10=150$

### Answer

$150$

### Exercise #4

$(15-9)\times(7-3)=$

### Step-by-Step Solution

According to the order of operations rules, we must first solve the expressions inside of the parentheses:

$15-9=6$

$7-3=4$

We obtain the following expression:

$6\times4=24$

### Answer

$24$

### Exercise #5

$[(5-2):3-1]\times4=$

### Step-by-Step Solution

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:

$0\times4=0$

### Answer

$0$

### Exercise #6

Choose the exercise for the highest result

### Step-by-Step Solution

Let's solve exercise A:

$8\times8=64$

Let's solve exercise B:

$10\times6=60$

Let's solve exercise C:

$12\times5=60$

Let's solve exercise D:

$15\times4=60$

### Answer

$8\times(12-4)$

### Exercise #7

$(16-6)\times9+(7-3)=$

### Step-by-Step Solution

According to the order of operations, we'll first solve the exercises in parentheses:

$(16-6)=10$

$(7-3)=4$

Now we'll get the exercise:

$10\times9+4$

We'll put the multiplication exercise in parentheses to avoid confusion in the rest of the solution:

$(10\times9)+4=$

According to the order of operations, we'll solve the multiplication exercise and then add:

$90+4=94$

### Answer

$94$

### Exercise #8

Solve the following exercise:

$4\cdot2-3:(1+3)=$

### Step-by-Step Solution

First, we solve the exercise within the parentheses:

$4\cdot2-3:4=$

We place multiplication and division exercises within parentheses:

$(4\cdot2)-(3:4)=$

We solve the exercises within the parentheses:

$8-\frac{3}{4}=7\frac{1}{4}$

### Answer

$7\frac{1}{4}$

### Exercise #9

$\big((5-4\cdot3)^2+8-3\big):2=$

### Step-by-Step Solution

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.

27

### Exercise #10

$12:3(1+1)=$

### Step-by-Step Solution

First, we perform the operation inside the parentheses:

$12:3(2)$

When there is no mathematical operation between parentheses and a number, we assume it is a multiplication.

Therefore, we can also write the exercise like this:

$12:3\times2$

Here we solve from left to right:

$12:3\times2=4\times2=8$

8

### Exercise #11

Solve the following:

$\frac{36-(4\cdot5)}{8}-3\cdot2=$

### Step-by-Step Solution

Let's first address the fraction. We must begin by solving the exercise within the parentheses due to the rules of the order of arithmetic operations. Parentheses come before everything else:

$\frac{36-(20)}{8}-3\times2=$

Let's continue by simplifying the fraction, we subtract the exercise in the numerator and divide by 8:

$\frac{36-20}{8}=\frac{16}{8}=2$

We then arrange the exercise accordingly:

$2-3\times2=$

Finally we solve the multiplication exercise and then subtract:

$2-6=-4$

-4

### Exercise #12

Solve the exercise:

$3:(4+5)\cdot9-6=$

### Step-by-Step Solution

We solve the exercise in parentheses:

$3:9\cdot9-6=$

$\frac{3}{9}\cdot9-6=$

We simplify and subtract:

$3-6=-3$

-3

### Exercise #13

Complete the following exercise:

$\frac{(5-3)\cdot15+3}{5+6}-\frac{2\cdot8}{3+1}=$

### Step-by-Step Solution

This simple example demonstrates the order of operations, which states that exponentiation precedes multiplication and division, which precede addition and subtraction, and that operations enclosed in parentheses precede all others,

Let's note that when a fraction (every fraction) is involved in a division operation, it means we can relate the numerator and the denominator to the fraction as whole numbers involved in multiplication, in other words, we can rewrite the given fraction and write it in the following form:

$\frac{(5-3)\cdot15+3}{5+6}-\frac{2\cdot8}{3+1}= \\ \downarrow\\ \big((5-3)\cdot15+3\big):(5+6)-(2\cdot8):(3+1)$We emphasize this by stating that fractions involved in the division and in their separate form , are actually found in multiplication,

Returning to the original fraction in the problem, in other words - in the given form, and simplifying, we separate the different fractions involved in the division operations and simplify them according to the order of operations mentioned, and in the given form:

$\frac{(5-3)\cdot15+3}{5+6}-\frac{2\cdot8}{3+1}= \\ \frac{2\cdot15+3}{11}-\frac{2\cdot8}{4}= \\ \frac{30+3}{11}-\frac{16}{4}=\\ \frac{33}{11}-\frac{16}{4}\\$In the first step, we simplified the fraction involved in the division from the left, in other words- we performed the multiplication operation in the division, in contrast, we performed the division operation involved in the fractions, in the next step we simplified the fraction involved in the division from the left and assumed that multiplication precedes division we started with the multiplication involved in this fraction and only then calculated the result of the division operation, in contrast, we performed the multiplication involved in the second division from the left,

We continue and simplify the fraction we received in the last step, this is done again according to the order of operations mentioned, in other words- we start with the division operation of the fractions (this is done by inverting the fractions) and in the next step calculate the result of the subtraction operation:

$\frac{33}{11}-\frac{16}{4}=\\ \frac{\not{33}}{\not{11}}-\frac{\not{16}}{\not{4}}=\\ 3-4=\\ -1$ We conclude the steps of simplifying the fraction, we found that:

$\frac{(5-3)\cdot15+3}{5+6}-\frac{2\cdot8}{3+1}= \\ \frac{2\cdot15+3}{11}-\frac{2\cdot8}{4}= \\ \frac{33}{11}-\frac{16}{4}=\\ 3-4=\\ -1$Therefore, the correct answer is answer d.

1-

### Exercise #14

$9-6:(4\times3)-1=$

### Step-by-Step Solution

We simplify this expression paying attention to the order of operations which states that exponentiation comes before multiplication and division, and before addition and subtraction, and that parentheses precede all of them.

Therefore, we start by performing the multiplication within parentheses, then we carry out the division operation, and we finish by performing the subtraction operation:

$9-6:(4\cdot3)-1= \\ 9-6:12-1= \\ 9-0.5-1= \\ 7.5$

Therefore, the correct answer is option C.

7.5

### Exercise #15

$(30+6):4\times3=$

### Step-by-Step Solution

According to the order of operations, first we solve the exercise within parentheses:

$30+6=36$

Now we solve the exercise

$36:4\times3=$

Since the exercise only involves multiplication and division operations, we solve from left to right:

$36:4=9$

$9\times3=27$

27