Parentheses - Examples, Exercises and Solutions

$8\times(5\times1)=$

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

$5\times1=5$

Now we multiply:

$8\times5=40$

40

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

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

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

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:

$0\times4=0$

$0$

$12:3(1+1)=$

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

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

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

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

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.

$1$

$(5\times4-10\times2)\times(3-5)=$

This simple rule is the order of operations which states that multiplication precedes addition and subtraction, and division precedes all of them,

In the given example, a multiplication occurs between two sets of parentheses, thus we simplify the expressions within each pair of parentheses separately,

We start with simplifying the expression within the parentheses on the left, this is done in accordance with the order of operations mentioned above, meaning that multiplication comes before subtraction, we perform the multiplications in this expression first and then proceed with the subtraction operations within it, in reverse we simplify the expression within the parentheses on the right and perform the subtraction operation within them:

What remains for us is to perform the last multiplication that was deferred, it is the multiplication that occurred between the expressions within the parentheses in the original expression, we perform it while remembering that multiplying any number by 0 will result in 0:

Therefore, the correct answer is answer d.

$0$

$(3+2-1):(1+3)-1+5=$

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.

$5$

Solve the exercise:

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

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:

$9+5=14$

$14$

Solve the following exercise:

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

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:

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

$7\frac{1}{4}$

Solve the exercise:

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

We solve the exercise in parentheses:

$3:9\cdot9-6=$

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

We simplify and subtract:

$3-6=-3$

-3

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

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:

$20-2=18$

$18$

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

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

Indicate whether the equality is true or not.

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

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,

So the correct answer is answer B.

Not true

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

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.

$-1$