# Parentheses - Examples, Exercises and Solutions

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.

1. Parentheses
2. Exponents and roots
3. Multiplications and divisions

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.

### Suggested Topics to Practice in Advance

1. Order of Operations: (Exponents)

## Practice Parentheses

### Exercise #1

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

### Step-by-Step Solution

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

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

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

99

### Exercise #2

$12:3(1+1)=$

### 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 #3

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

### 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 #4

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

### 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:

$0\times4=0$

$0$

### Exercise #5

$20-(1+9:9)=$

### Step-by-Step Solution

First, we solve the exercise in the parentheses

$(1+9:9)=$

According to the order of operations, we first divide and then add:

$1+1=2$

Now we obtain the exercise:

$20-2=18$

$18$

### Exercise #1

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

### 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

### Exercise #2

Solve the exercise:

$3\cdot(4-1)+5:1=$

### Step-by-Step Solution

We solve the exercise in parentheses:$3\cdot3+5:1=$

We place in parentheses the multiplication and division exercises:

$(3\cdot3)+(5:1)=$

We solve the exercises in parentheses:

$9+5=14$

$14$

### Exercise #3

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:

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

$7\frac{1}{4}$

### Exercise #4

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 #5

Indicate whether the equality is true or not.

$5^3:(4^2+3^2)-(\sqrt{100}-8^2)=5^3:4^2+3^2-\sqrt{100}+8^2$

### Step-by-Step Solution

To determine if the given equality is correct we will simplify each of the expressions that appear in it separately,

This is done while keeping in mind the order of operations which states that multiplication precedes division and subtraction precedes addition and that parentheses precede all,

A. Let's start then with the expression on the left side of the given equality:

$5^3:(4^2+3^2)-(\sqrt{100}-8^2)$We start by simplifying the expressions inside the parentheses, this is done by calculating their numerical value (while remembering the definition of the square root as the non-negative number whose square gives the number under the root), in parallel we calculate the numerical value of the other terms in the expressions:

$5^3:(4^2+3^2)-(\sqrt{100}-8^2) =\\ 125:(16+9)-(10-64)$We continue and finish simplifying the expressions inside the parentheses, meaning we perform the subtraction operation in them, then we perform the division operation which is in the first term from the left and then the remaining subtraction operation:

$125:(16+9)-(10-64) =\\ 125:25-(-54) =\\ 5+54 = 59$We note that the result of the subtraction operation in the parentheses is a negative result and therefore in the next step we will leave this result in the parentheses and then apply the multiplication law which states that multiplying a negative number by a negative number will give a positive result (so that in the end an addition operation is obtained), then, we perform the addition operation in the expression that was obtained,

We finished simplifying the expression on the left side of the given equality, let's summarize the simplification steps:

$5^3:(4^2+3^2)-(\sqrt{100}-8^2) =\\ 125:(16+9)-(10-64) =\\ 5+54 =\\ 59$

B. We continue from simplifying the expression on the right side of the given equality:

$5^3:4^2+3^2-\sqrt{100}+8^2$We recall again the order of operations which states that multiplication precedes division and subtraction precedes addition and that parentheses precede all, and note that although in this expression there are no parentheses, there are terms in fractions and a division operation, so we start by calculating their numerical value, then we perform the division operation:

$5^3:4^2+3^2-\sqrt{100}+8^2 =\\ 125:16+9-10+64 =\\ 7\frac{13}{16}+9-10+64=\\ 70\frac{13}{16}$We note that since the division operation that was performed in the first term from the left yielded an incomplete result (greater than the divisor), we marked this result as a mixed number, then we performed the remaining addition and subtraction operations,

We finished simplifying the expression on the right side of the given equality, the simplification of this expression is short, so there is no need to summarize,

Let's go back now to the given equality and place in it the results of simplifying the expressions that were detailed in A and B:

$5^3:(4^2+3^2)-(\sqrt{100}-8^2)=5^3:4^2+3^2-\sqrt{100}+8^2 \\ \downarrow\\ 59= 70\frac{13}{16}$As can be seen this equality does not hold, meaning - we got a false sentence,

Not true

### Exercise #1

$[(27:3)-9\cdot2]+(5+3)=$

### Step-by-Step Solution

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

Let's keep in mind that in the expression of the problem there are no parentheses or powers, but there are multiplication and division operations, so we start with them, later 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.

$-1$

### Exercise #2

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

### Step-by-Step Solution

This expression is simplified while maintaining the order of operations which states that parentheses take precedence over exponents and the previous exponents take precedence over multiplication and division and that the previous ones take precedence over all,

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 them and then we will perform the exponentiation on them, then we will deal similarly with the outer parentheses where we will simplify the expression within them, all of the above we will perform while maintaining the order of operations specified at the beginning of the solution:

$\big((5-4\cdot3)^2+8-3\big):2= \\ \big((5-12)^2+8-3\big):2= \\ \big((-7)^2+8-3\big):2= \\ \big(49+8-3\big):2=\\$Note that since exponents and division precede multiplication and division we first performed the exponentiation in the inner parentheses and then the division operation in them, we continued and performed first the exponentiation of the expression results in the inner parentheses by squaring, this is while we remember that raising any number (positive or negative) to an even power will always give a positive result, then we performed the remaining multiplication and division operations in the outer parentheses that remained, emphasizing again that exponentiation precedes the other operations within the outer parentheses, i.e. both for multiplication and division and for exponents and division (if any),

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 #3

Solve the following:

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

### Step-by-Step Solution

Let's start by solving the fraction, and we solve the exercise within the parentheses since, according to the rules of the order of arithmetic operations, parentheses come before everything else:

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

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

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

We arrange the exercise accordingly:

$2-3\times2=$

We solve the multiplication exercise and then subtract:

$2-6=-4$

-4

### Exercise #4

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 #5

$(3\times5-15\times1)+3-2=$

### Step-by-Step Solution

This simple rule is 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,

Following the simple rule, multiplication comes before division and subtraction, therefore we calculate the values of the multiplications and then proceed with the operations of division and subtraction

$3\cdot5-15\cdot1+3-2= \\ 15-15+3-2= \\ 1$ Therefore, the correct answer is answer B.

$1$