Additional Arithmetic Rules: Number of terms

$7-(10-(4-3))=$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

$7-(10-(4-3))=$
In this exercise, there are only subtraction operations and parentheses within parentheses.

Therefore, we will first perform the operation of the inner parentheses. After which we will be able to remove the inner parentheses and will be left with only one pair of parentheses.

$7-(10-(1))=7-(10-1)=$

When we perform an operation of ("addition and subtraction") with directed numbers, we enclose the directed number inside of parentheses.

Parentheses can be omitted, but when omitting the parentheses, remember that (-=+-)

$7-(9)=7-9=-2$

Reminder: addition and subtraction of directed numbers

In this case, we remember that when we have two numbers with different signs, it is important to determine which number is greater in terms of absolute value (absolute - the distance from zero). The larger number will determine the sign of the result and we will perform a subtraction exercise.

Therefore, the correct answer is option c: (-2)

$-2$

$75:(8:(4:15))=\text{?}$

First, we rewrite the multiplication exercise within the inner parentheses as a fraction:

$75:(8:\frac{4}{15})=$

Now we invert the fraction to create a multiplication exercise:

$75:(8\times\frac{15}{4})=$

We add the 8 to the numerator of the fraction in the multiplication exercise:

$75:\frac{8\times15}{4}=$

Now we invert the fraction to create a multiplication exercise:

$75\times\frac{4}{8\times15}=$

We add the 75 to the numerator of the fraction in the multiplication exercise:

$\frac{75\times4}{8\times15}=$

We break down the 75 and the 8 into smaller multiplication exercises:

$\frac{15\times5\times4}{4\times2\times15}=$

We simplify the 4 and the 15 in the numerator and denominator:

$\frac{5}{2}=2\frac{1}{2}$

$2\frac{1}{2}$

$15/(4/(2:8))=\text{?}$

First, we rewrite the multiplication exercise inside parentheses as a fraction:

$15/(4:\frac{2}{8})=$

Now we invert the fraction to create a multiplication exercise:

$15/(4\times\frac{8}{2})=$

We add the 4 to the numerator of the fraction in the multiplication exercise:

$15/(\frac{4\times8}{2})=$

Now we invert the fraction to create a multiplication exercise:

$15\times\frac{2}{4\times8}=$

We add the 15 to the numerator of the fraction in the multiplication exercise:

$\frac{15\times2}{4\times8}=$

We separate the 4 into a smaller multiplication exercise:

$\frac{15\times2}{2\times2\times8}=$

We simplify the 2 in the numerator and denominator:

$\frac{15}{2\times8}=\frac{15}{16}$

$\frac{15}{16}$

$3:(4:(5:12))=\text{?}$

First, we rewrite the multiplication exercise inside parentheses as a fraction:

$3:(4:\frac{5}{12})=$

Now we invert the fraction to create a multiplication exercise:

$3:(4\times\frac{12}{5})=$

We add the 4 to the numerator of the fraction in the multiplication exercise:

$3:\frac{4\times12}{5}=$

Now we invert the fraction to create a multiplication exercise:

$3\times\frac{5}{4\times12}=$

We add the 3 to the numerator of the fraction in the multiplication exercise:

$\frac{3\times5}{4\times12}=$

We break down the 12 into a smaller multiplication exercise:

$\frac{3\times5}{4\times3\times4}=$

We simplify the 3 in the numerator and denominator:

$\frac{5}{4\times4}=\frac{5}{16}$

$\frac{5}{16}$

$10-(12-(4-12))=$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

$10-(12-(4-12))=$

In this exercise, there are only subtraction operations and parentheses within parentheses.

Therefore, we will first perform the operation of the inner parentheses. After which we will be able to remove the inner parentheses and will be left with only one pair of parentheses.

$10-(12-(4-12))=10-(12-(-8))$

Keep in mind

What happens when there is a sequence of two signs present in the exercise (these are usually separated by parentheses) Let's highlight several different cases:

When a sequence with two plus signs appears, the result will also be a plus.

When a sequence with two minus signs appears, the result will also be a plus.

When a sequence with minus and plus signs or plus and minus signs appears, the result will be minus.

$10-(12-(-8)) = 10-(12+8)$

Now we will perform the operation of the remaining parentheses and after the calculation, we will remove the parentheses.

$10-(12+8)=10-(20)$$10-\left(20\right)=10-20=-10$

Therefore, the correct answer is option a: (-10)

$-10$

$17-(3-(-7-4))=$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

$17-(3-(-7-4))=$

In this exercise, there are only subtraction operations and parentheses within parentheses.

Therefore, we will first perform the operation of the inner parentheses. After which we will be able to remove the inner parentheses and will be left with only one pair of parentheses.

Adding and subtracting directed numbers is based on several key principles. In this case, we will remember that when we have two directed numbers with the same sign (plus or minus), the result will also have the same sign, which will actually be the result of an addition exercise.

$17-(3-(-7-4))= 17-(3-(-11))$
Keep in mind

What happens when there is a sequence of two signs present in the exercise (these are usually separated by parentheses) Let's highlight several different cases:

When a sequence with two plus signs appears, the result will also be a plus.

When a sequence with two minus signs appears, the result will also be a plus.

When a sequence with minus and plus signs or plus and minus signs appears, the result will be minus.

$17-(3-(-11))=17-(3+11)=$
Now we will perform the operation of the remaining parentheses and after the calculation, we will remove the parentheses.

$17-(14)=17-14=3$

Therefore, the correct answer is option b: (3)

$3$

$10:(2:(15:7))=\text{?}$

First, we rewrite the multiplication exercise inside parentheses as a fraction:

$10:(2:\frac{15}{7})=$

Now we invert the fraction to create a multiplication exercise:

$10:(2\times\frac{7}{15})=$

We add the 2 to the numerator of the fraction in the multiplication exercise:

$10:\frac{2\times7}{15}=$

Now we invert the fraction to create a multiplication exercise:

$10\times\frac{15}{2\times7}=$

We add the 10 to the numerator of the fraction in the multiplication exercise:

$\frac{10\times15}{2\times7}=$

We break down the 10 into a smaller multiplication exercise.

$\frac{5\times2\times15}{2\times7}=$

We simplify the 2 in the numerator and denominator:

$\frac{5\times15}{7}=\frac{75}{7}$

We separate the fraction into a sum exercise between fractions:

$\frac{70+5}{7}=\frac{70}{7}+\frac{5}{7}=10+\frac{5}{7}=10\frac{5}{7}$

$10\frac{5}{7}$

$13-(10-((-4)-(-10)))=$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

$13-(10-((-4)-(-10))) =$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

Keep in mind

What happens when there is a sequence of two signs present in the exercise (these are usually separated by parentheses) Let's highlight several different cases:

When a sequence with two plus signs appears, the result will also be a plus.

When a sequence with two minus signs appears, the result will also be a plus.

When a sequence with minus and plus signs or plus and minus signs appears, the result will be minus.

$13-(10-((-4)-(-10))) = 13-(10-((-4)+10))$

Reminder: addition and subtraction of directed numbers

When we have two numbers with different signs, it is important to determine which number is greater in terms of absolute value (absolute - the distance from zero). The larger number will determine the sign of the result and we will perform a subtraction exercise.

Now we will perform the operation of the remaining parentheses and after the calculation, we will remove the parentheses.

$13-(4) = 13-4 = 9$

Therefore, the correct answer is c: (9)

$9$

$49:(7:(\frac{4}{3}:7))=\text{?}$

First, we rewrite the multiplication exercise in the innermost parentheses as a fraction:

$49:(7:(\frac{4}{3}\times\frac{1}{7}))=$

We multiply the fractions and combine them since it is just a multiplication operation:

$49:(7:\frac{4\times1}{3\times7})=$

Now we invert the fraction to create a multiplication exercise:

$49:(7\times\frac{3\times7}{4\times1})=$

We add the 7 to the numerator of the fraction in the multiplication exercise:

$49:\frac{7\times3\times7}{4\times1}=$

We invert the fraction to create a multiplication exercise:

$49\times\frac{4\times1}{7\times3\times7}=$

We add the 49 to the numerator of the fraction in the multiplication exercise:

$\frac{49\times4\times1}{7\times3\times7}=$

We break down the 49 into a smaller multiplication exercise:

$\frac{7\times7\times4\times1}{7\times3\times7}=$

We simplify the 7 in the numerator and denominator:

$\frac{4\times1}{3}=\frac{4}{3}=1\frac{1}{3}$

$1\frac{1}{3}$

$74-(78-(10-13))=$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

$74-(78-(10-13)=$

In this exercise, there are only subtraction operations and parentheses within parentheses.

Therefore, we will first perform the operation of the inner parentheses. After which we will be able to remove the inner parentheses and will be left with only one pair of parentheses.

$74-(78-(10-13)= 74-(78-(-3))$

Keep in mind

What happens when there is a sequence of two signs present in the exercise (these are usually separated by parentheses) Let's highlight several different cases:

When a sequence with two plus signs appears, the result will also be a plus.

When a sequence with two minus signs appears, the result will also be a plus.

When a sequence with minus and plus signs or plus and minus signs appears, the result will be minus.

$74-(78-(-3)) =74-(78+3)$
Now we will perform the operation of the remaining parentheses and after the calculation we will remove the parentheses entirely.

$74-(78+3)=74-(81)$$74-(81)=74-81=-7$

Therefore, the answer is option b: (-7)

$-7$

$25:(3:(10:35))=\text{?}$

First, we rewrite down the multiplication exercise within the inner parentheses as a fraction:$25:(3:\frac{10}{35})=$

Now we invert the fraction to create a multiplication exercise:

$25:(3\times\frac{35}{10})=$

We add the 3 to the numerator of the fraction in the multiplication exercise:

$25:\frac{3\times35}{10}=$

Now we invert the fraction to create a multiplication exercise:

$25\times\frac{10}{3\times35}=$

We add the 25 to the numerator of the fraction in the multiplication exercise:

$\frac{25\times10}{3\times35}=$

We separate the 10 and 35 into smaller multiplication exercises:

$\frac{25\times5\times2}{3\times7\times5}=$

We simplify the 5 in the numerator and denominator:

$\frac{25\times2}{3\times7}=\frac{50}{21}$

We separate the numerator of the fraction into a sum exercise:

$\frac{42+8}{21}=$

We separate into sum exercises among fractions:

$\frac{42+8}{21}=\frac{42}{21}+\frac{8}{21}=2+\frac{8}{21}=2\frac{8}{21}$

$2\frac{8}{21}$

$25-(34-(20-8))=$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

$25-(34-(20-8))=$

In this exercise, there are only subtraction operations and parentheses within parentheses.

Therefore, we will first perform the operation of the inner parentheses. After which we will be able to remove the inner parentheses and will be left with only one pair of parentheses.

$25-(34-(20-8))=25-(34-(12))$
Keep in mind

What happens when there is a sequence of two signs present in the exercise (these are usually separated by parentheses) Let's highlight several different cases:

When a sequence with two plus signs appears, the result will also be a plus.

When a sequence with two minus signs appears, the result will also be a plus.

When a sequence with minus and plus signs or plus and minus signs appears, the result will be minus.

$25-(34-(12))=25-(34-12)$
Now we will perform the operation of the remaining parentheses and after the calculation, we will remove the parentheses.

$25-(34-12)=25-(22)$

$25-(22)=25-22=3$

Therefore, the correct answer is option c: (3)

$3$

$\frac{2}{7}:(\frac{3}{5}:(7:8))=\text{?}$

First, we rewrite the multiplication exercise inside parentheses as a fraction:

$\frac{2}{7}:(\frac{3}{5}:\frac{7}{8})=$

Now we invert the fraction to create a multiplication exercise:

$\frac{2}{7}:(\frac{3}{5}\times\frac{8}{7})=\frac{2}{7}:(\frac{3\times8}{5\times7})$

Now we invert the fraction to create a multiplication exercise:

$\frac{2}{7}\times\frac{5\times7}{3\times8}=\frac{2\times5\times7}{7\times3\times8}$

We simplify the 7 in the numerator and denominator:

$\frac{2\times5}{3\times8}=$

We break down the 8 into a shorter multiplication exercise:

$\frac{2\times5}{3\times2\times4}=$

We simplify the 2 in the numerator and denominator:

$\frac{5}{3\times4}=\frac{5}{12}$

$\frac{5}{12}$

$10/(7/(9/2))=\text{?}$

We rewrite the innermost parentheses in fraction form:

$10/(7:\frac{9}{2})=$

We convert the parentheses into a multiplication exercise through inverting the fraction:

$10/(7\times\frac{2}{9})=$

We add the 7 to the numerator for the multiplication exercise:

$10/(\frac{7\times2}{9})=10:\frac{7\times2}{9}$

We convert the exercise into a multiplication by inverting the fraction:

$10\times\frac{9}{7\times2}=$

We add the 10 to the numerator for the multiplication exercise:

$\frac{10\times9}{7\times2}=$

We break down the 10 into a simpler multiplication exercise:

$\frac{5\times2\times9}{7\times2}=$

We simplify the 2 in the numerator and denominator:

$\frac{5\times9}{7}=\frac{45}{7}$

We convert the fraction's numerator into a sum exercise:

$\frac{42+3}{7}=\frac{42}{7}+\frac{3}{7}=6+\frac{3}{7}=6\frac{3}{7}$

$6\frac{3}{7}$

$12/(7\cdot(4/(3\cdot(12/(3\cdot2)))))=\text{?}$

First, we start with the innermost parentheses, we rewrite the exercise in fraction form:

$12/(7\cdot(4/(3\cdot(\frac{12}{3\times2})))=$

We associate the 3 in the innermost parenthesis with the multiplication exercise in the numerator and convert the division exerciseinto a multiplication exercise in the next parenthesis:

$12/(7\cdot(4\times(\frac{3\times2}{3\times12}))=$

We add the 4 to the multiplication exercise in the numerator of the fraction:

$12/(7\cdot(\frac{4\times3\times2}{3\times12})=$

We add the 7 to the multiplication exercise in the numerator of the fraction:

$12/(\frac{7\times4\times3\times2}{3\times12})=$

We convert the exercise into a multiplication by inverting the fraction between numerator and denominator:

$12\times\frac{3\times12}{7\times4\times3\times2}=$

We add the 12 to the multiplication exercise in the numerator of the fraction:

$\frac{12\times3\times12}{7\times4\times3\times2}=$

We simplify the 3 in the numerator and denominator, and break down the 12 in the numerator for a simpler multiplication exercise:

$\frac{4\times3\times2\times6}{7\times4\times2}=$

We simplify to 4 and 2 in the numerator and denominator:

$\frac{3\times6}{7}=\frac{18}{7}$

We break down the fraction into a sum exercise:

$\frac{14+4}{7}=\frac{14}{7}+\frac{4}{7}=2+\frac{4}{7}=2\frac{4}{7}$

$2\frac{4}{7}$

$-30-((-41)-((-4)-(-8)))=$

Let's begin by remembering the order of arithmetic operations: calculation of the operation within parentheses, multiplication and division (from left to right), addition and subtraction (from left to right)

We must also underline that when there are parentheses within parentheses, we should start with the innermost first.

$-30-((-41)-((-4)-(-8))) =$In this exercise, there are only subtraction operations and parentheses within parentheses.

Therefore, we will first perform the operation of the inner parentheses. After which we will be able to remove the inner parentheses and will be left with only one pair of parentheses.

Keep in mind

What happens when there is a sequence of two signs present in the exercise (these are usually separated by parentheses) Let's highlight several different cases:

When a sequence with two plus signs appears, the result will also be a plus.

When a sequence with two minus signs appears, the result will also be a plus.

When a sequence with minus and plus signs or plus and minus signs appears, the result will be minus.

$-30-((-41)-((-4)-(-8))) = -30-((-41)-((-4)+8))$

Reminder: addition and subtraction of directed numbers

When we have two numbers with different signs, it is important to determine which number is greater in terms of absolute value (absolute - the distance from zero). The larger number will determine the sign of the result and we will perform a subtraction exercise.

$-30-((-41)-((-4)+8)) = -30-((-41)-(4))$
$-30-((-41)-(4)) = -30-(-41-4)$

Now we will perform the operation of the remaining parentheses and after the calculation, we will remove the parentheses.

$-30-(-41-4) = -30-(-45)$

$-30-(-45) = -30+45 = 15$

Therefore, the correct answer is option b: (15)

$15$

$2/(8\cdot(4/(3\cdot(10/(4\cdot8)))))=?$

First, we will start with the innermost parentheses, we rewrite the expression in fraction form:

$2/(8\cdot(4/(3\cdot\frac{10}{4\times8})))=$

We add the 3 to the numerator of the fraction and create a multiplication expression:

$2/(8\cdot(4/(\frac{3\times10}{4\times8})))=$

Again, we address the innermost parentheses, we invert the fraction to create a multiplication expression:

$2/(8\cdot(4\times\frac{4\times8}{3\times10}))=$

We add the 4 to the numerator of the fraction and create a multiplication exercise:

$2/(8\cdot\frac{4\times4\times8}{3\times10})=$

We add the 8 to the numerator of the fraction and create a multiplication exercise:

$2:\frac{8\times4\times4\times8}{3\times10}=$

Now we will invert the fraction to create a multiplication exercise:

$2\times\frac{3\times10}{8\times4\times4\times8}=$

We add the 2 to the numerator of the fraction and create a multiplication exercise:

$\frac{2\times3\times10}{8\times4\times4\times8}=$

We break down the 10 in the numerator and the 4 in the denominator into smaller multiplication exercises:

$\frac{2\times3\times2\times5}{8\times2\times2\times4\times8}=$

We simplify to 2 in the numerator and denominator:

$\frac{3\times5}{8\times4\times8}=$

We solve the multiplication exercises in the numerator and denominator from left to right:

$\frac{3\times5}{8\times4\times8}=\frac{15}{32\times8}=\frac{15}{256}$

$\frac{15}{256}$

$\frac{15}{3}:(\frac{12}{8}:(1\frac{4}{5}:8))=\text{?}$

We solve the first fraction exercise:

$\frac{15}{3}=5$

We solve the second fraction by breaking it down into smaller multiplication exercises:

$\frac{12}{8}=\frac{4\times3}{4\times2}=\frac{3}{2}$

We convert the third fraction into a simple fraction:

$1\frac{4}{5}=\frac{9}{5}$

Now we obtain the exercise:

$5:(\frac{3}{2}:(\frac{9}{5}:8))=$

We note in inner parentheses the division exercise, a multiplication exercise between fractions:

$5:(\frac{3}{2}:(\frac{9}{5}\times\frac{1}{8}))=$

$5:(\frac{3}{2}:\frac{9}{5\times8})=$

Now we invert the fraction to create a multiplication exercise:

$5:(\frac{3}{2}\times\frac{5\times8}{9})=$

We combine the multiplication exercises, since it's just a multiplication operation:

$5:\frac{3\times5\times8}{2\times9}=$

Now we invert the fraction to create a multiplication exercise:

$5\times\frac{2\times9}{3\times5\times8}=$

We add the 5 to the numerator of the fraction in the multiplication exercise:

$\frac{5\times2\times9}{3\times5\times8}=$

We simplify the 5 in the numerator and denominator:

$\frac{2\times9}{3\times8}=$

We break down the 8 into a smaller multiplication exercise:

$\frac{2\times9}{3\times2\times4}=$

We simplify the 2 in the numerator and denominator:

$\frac{9}{3\times4}=$

We break down the 9 into a smaller multiplication exercise:

$\frac{3\times3}{3\times4}=$

We simplify the 3 in the numerator and denominator:

$\frac{3}{4}$

$\frac{3}{4}$

$43-(35-(24-18))=$

$14$

$7:(8\times(10:(3\times12)))=$

$3.15$