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Use the Language of Algebra

Use variables and algebraic symbols

Use Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is \(20\) years old and Alex is \(23,\) so Alex is \(3\) years older than Greg. When Greg was \(12,\) Alex was \(15.\) When Greg is \(35,\) Alex will be \(38.\) No matter what Greg’s age is, Alex’s age will always be \(3\) years more, right?

In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The \(3\) years between them always stays the same, so the age difference is the constant.

In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age \(g.\) Then we could use \(g+3\) to represent Alex’s age. See .

Greg’s age	Alex’s age
\(12\)	\(15\)
\(20\)	\(23\)
\(35\)	\(38\)
\(g\)	\(g+3\)

Letters are used to represent variables. Letters often used for variables are \(x,y,a,b,\text{and}\ c.\)

To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. In Whole Numbers, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.

Operation	Notation	Say:	The result is…
Addition	\(a+b\)	\(a\ \text{plus}\ b\)	the sum of \(a\) and \(b\)
Subtraction	\(a-b\)	\(a\ \text{minus}\ b\)	the difference of \(a\) and \(b\)
Multiplication	\(a\cdot b,(a)(b),(a)b,a(b)\)	\(a\ \text{times}\ b\)	The product of \(a\) and \(b\)
Division	\(a\div b,a/b,\ \frac{a}{b},ba\)	\(a\) divided by \(b\)	The quotient of \(a\) and \(b\)

In algebra, the cross symbol, \(\times ,\) is not used to show multiplication because that symbol may cause confusion. Does \(3xy\) mean \(3\ \times \ y\) (three times \(y\)) or \(3\cdot x\cdot y\) (three times \(x\ \text{times}\ y\))? To make it clear, use • or parentheses for multiplication.

We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.

The sum of \(5\) and \(3\) means add \(5\) plus \(3,\) which we write as \(5+3.\)
The difference of \(9\) and \(2\) means subtract \(9\) minus \(2,\) which we write as \(9-2.\)
The product of \(4\) and \(8\) means multiply \(4\) times \(8,\) which we can write as \(4\cdot 8.\)
The quotient of \(20\) and \(5\) means divide \(20\) by \(5,\) which we can write as \(20\div 5.\)

Common Grouping Symbols
parentheses	\((\ )\)
brackets	\([\ ]\)
braces	\(\{\ \}\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Identify Expressions and Equations

What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb.

In algebra, we have expressions and equations. An expression is like a phrase. Here are some examples of expressions and how they relate to word phrases:

Expression	Words	Phrase
\(3+5\)	\(3\ \text{plus}\ 5\)	the sum of three and five
\(n-1\)	\(n\) minus one	the difference of \(n\) and one
\(6\cdot 7\)	\(6\ \text{times}\ 7\)	the product of six and seven
\(\frac{x}{y}\)	\(x\) divided by \(y\)	the quotient of \(x\) and \(y\)

Notice that the phrases do not form a complete sentence because the phrase does not have a verb. An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb. Here are some examples of equations:

Equation	Sentence
\(3+5=8\)	The sum of three and five is equal to eight.
\(n-1=14\)	\(n\) minus one equals fourteen.
\(6\cdot 7=42\)	The product of six and seven is equal to forty-two.
\(x=53\)	\(x\) is equal to fifty-three.
\(y+9=2y-3\)	\(y\) plus nine is equal to two \(y\) minus three.

Example

Try it.

Determine if each is an expression or an equation:

ⓐ \(\ 16-6=10\)
ⓑ \(\ 4\cdot 2+1\)
ⓒ \(\ x\div 25\)
ⓓ \(\ y+8=40\)

Solution

ⓐ \(\ 16-6=10\)	This is an equation—two expressions are connected with an equal sign.
ⓑ \(\ 4\cdot 2+1\)	This is an expression—no equal sign.
ⓒ \(\ x\div 25\)	This is an expression—no equal sign.
ⓓ \(\ y+8=40\)	This is an equation—two expressions are connected with an equal sign.

Simplify Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify \(4\cdot 2+1\) we’d first multiply \(4\cdot 2\) to get \(8\) and then add the \(1\) to get \(9.\) A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:

\[4\cdot 2+1\]\[8+1\]\[9\]

Suppose we have the expression \(2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2.\) We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write \(2\cdot 2\cdot 2\) as \({2}^{3}\) and \(2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\) as \({2}^{9}.\) In expressions such as \({2}^{3},\) the \(2\) is called the base and the \(3\) is called the exponent. The exponent tells us how many factors of the base we have to multiply.

\[\text{means multiply three factors of 2}\]

We say \({2}^{3}\) is in exponential notation and \(2\cdot 2\cdot 2\) is in expanded notation.

For powers of \(n=2\) and \(n=3,\) we have special names.

\[\begin{array}{l}{a}^{2}\ \text{is read as}\ \text{"}a\ \text{squared"} \\ {a}^{3}\ \text{is read as}\ \text{"}a\ \text{cubed"}\end{array}\]

lists some examples of expressions written in exponential notation.

Exponential Notation	In Words
\({7}^{2}\)	\(7\) to the second power, or \(7\) squared
\({5}^{3}\)	\(5\) to the third power, or \(5\) cubed
\({9}^{4}\)	\(9\) to the fourth power
\({12}^{5}\)	\(12\) to the fifth power

Example

Try it.

Write each expression in exponential form:

ⓐ \(\ 16\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16\)
ⓑ \(\ 9\cdot 9\cdot 9\cdot 9\cdot 9\)
ⓒ \(\ x\cdot x\cdot x\cdot x\)
ⓓ \(\ a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\)

Solution

ⓐ The base 16 is a factor 7 times.	\({16}^{7}\)
ⓑ The base 9 is a factor 5 times.	\({9}^{5}\)
ⓒ The base \(x\) is a factor 4 times.	\({x}^{4}\)
ⓓ The base \(a\) is a factor 8 times.	\({a}^{8}\)

Example

Try it.

Write each exponential expression in expanded form:

ⓐ \(\ {8}^{6}\\)
ⓑ \(\ {x}^{5}\)

Solution

ⓐ The base is \(8\) and the exponent is \(6,\) so \({8}^{6}\) means \(8\cdot 8\cdot 8\cdot 8\cdot 8\cdot 8\)

ⓑ The base is \(x\) and the exponent is \(5,\) so \({x}^{5}\) means \(x\cdot x\cdot x\cdot x\cdot x\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Simplify Expressions Using the Order of Operations

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values.

For example, consider the expression:

\[4+3\cdot 7\]\[\begin{array}{llllllllllllllll}\text{Some students say it simplifies to 49.} & \ & & \text{Some students say it simplifies to 25.} \\ \begin{array}{lll} & & 4+3\cdot 7 \\ \text{Since}\ 4+3\ \text{gives 7.} & \ & 7\cdot 7 \\ \text{And}\ 7\cdot 7\ \text{is 49.} & \ & 49\end{array} & & & \begin{array}{lll} & & 4+3\cdot 7 \\ \text{Since}\ 3\cdot 7\ \text{is 21.} & & 4+21 \\ \text{And}\ 21+4\ \text{makes 25.} & & 25\end{array}\end{array}\]

Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.

Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase. Please Excuse My Dear Aunt Sally.

Order of Operations
Please	Parentheses
Excuse	Exponents
My Dear	Multiplication and Division
Aunt Sally	Addition and Subtraction

It’s good that ‘My Dear’ goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.

Similarly, ‘Aunt Sally’ goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Operation	Notation	Say:	The result is…
Addition	\(a+b\)	\(a\ \text{plus}\ b\)	the sum of \(a\) and \(b\)
Multiplication	\(a\cdot b,(a)(b),(a)b,a(b)\)	\(a\ \text{times}\ b\)	The product of \(a\) and \(b\)
Subtraction	\(a-b\)	\(a\ \text{minus}\ b\)	the difference of \(a\) and \(b\)
Division	\(a\div b,a/b,\ \frac{a}{b},ba\)	\(a\) divided by \(b\)	The quotient of \(a\) and \(b\)

Equality Symbol
- \(a=b\) is read as \(a\) is equal to \(b\)
- The symbol \(=\) is called the equal sign.
Inequality
- \(a
- \(a\) is to the left of \(b\) on the number line
- \(a>b\) is read \(a\) is greater than \(b\)
- \(a\) is to the right of \(b\) on the number line

Algebraic Notation	Say
\(a=b\)	\(a\) is equal to \(b\)
\(a\ne b\)	\(a\) is not equal to \(b\)
\(a	\(a\) is less than \(b\)
\(a>b\)	\(a\) is greater than \(b\)
\(a\le b\)	\(a\) is less than or equal to \(b\)
\(a\ge b\)	\(a\) is greater than or equal to \(b\)

Exponential Notation
- For any expression \({a}^{n}\) is a factor multiplied by itself \(n\) times, if \(n\) is a positive integer.
- \({a}^{n}\) means multiply \(n\) factors of \(a\)
- The expression of \({a}^{n}\) is read \(a\) to the \(n\text{th}\) power.

Order of Operations When simplifying mathematical expressions perform the operations in the following order:

Parentheses and other Grouping Symbols: Simplify all expressions inside the parentheses or other grouping symbols, working on the innermost parentheses first.
Exponents: Simplify all expressions with exponents.
Multiplication and Division: Perform all multiplication and division in order from left to right. These operations have equal priority.
Addition and Subtraction: Perform all addition and subtraction in order from left to right. These operations have equal priority.

Use the Language of Algebra

Use Variables and Algebraic Symbols

In the following exercises, translate from algebraic notation to words.

Try it.

\(16-9\)

Solution

16 minus 9, the difference of sixteen and nine

Try it.

\(25-7\)

Try it.

\(5\cdot 6\)

Solution

5 times 6, the product of five and six

Try it.

\(3\cdot 9\)

Try it.

\(28\div 4\)

Solution

28 divided by 4, the quotient of twenty-eight and four

Try it.

\(45\div 5\)

Try it.

\(x+8\)

Solution

x plus 8, the sum of x and eight

Try it.

\(x+11\)

Try it.

\((2)(7)\)

Solution

2 times 7, the product of two and seven

Try it.

\((4)(8)\)

Try it.

\(14<21\)

Solution

fourteen is less than twenty-one

Try it.

\(17<35\)

Try it.

\(36\ge 19\)

Solution

thirty-six is greater than or equal to nineteen

Try it.

\(42\ge 27\)

Try it.

\(3n=24\)

Solution

3 times n equals 24, the product of three and n equals twenty-four

Try it.

\(6n=36\)

Try it.

\(y-1>6\)

Solution

y minus 1 is greater than 6, the difference of y and one is greater than six

Try it.

\(y-4>8\)

Try it.

\(2\le 18\div 6\)

Solution

2 is less than or equal to 18 divided by 6; 2 is less than or equal to the quotient of eighteen and six

Try it.

\(3\le 20\div 4\)

Try it.

\(a\ne 7\cdot 4\)

Solution

a is not equal to 7 times 4, a is not equal to the product of seven and four

Try it.

\(a\ne 1\cdot 12\)

Identify Expressions and Equations

In the following exercises, determine if each is an expression or an equation.

Try it.

\(9\cdot 6=54\)

Solution

equation

Try it.

\(7\cdot 9=63\)

Try it.

\(5\cdot 4+3\)

Solution

expression

Try it.

\(6\cdot 3+5\)

Try it.

\(x+7\)

Solution

expression

Try it.

\(x+9\)

Try it.

\(y-5=25\)

Solution

equation

Try it.

\(y-8=32\)

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

Try it.

\(3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\cdot 3\)

Solution

3⁷

Try it.

\(4\cdot 4\cdot 4\cdot 4\cdot 4\cdot 4\)

Try it.

\(x\cdot x\cdot x\cdot x\cdot x\)

Solution

x⁵

Try it.

\(y\cdot y\cdot y\cdot y\cdot y\cdot y\)

In the following exercises, write in expanded form.

Try it.

\({5}^{3}\)

Solution

\(5\cdot 5\cdot 5\)

Try it.

\({8}^{3}\)

Try it.

\({2}^{8}\)

Solution

\(2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\)

Try it.

\({10}^{5}\)

Simplify Expressions Using the Order of Operations

In the following exercises, simplify.

Try it.

ⓐ \(\ 3+8\cdot 5\ \\)
ⓑ \(\ \text{(3+8)}\cdot 5\)

Solution

ⓐ 43
ⓑ 55

Try it.

ⓐ \(\ 2+6\cdot 3\ \\)
ⓑ \(\ \text{(2+6)}\cdot 3\)

Try it.

\({2}^{3}-12\div (9-5)\)

Solution

Try it.

\({3}^{2}-18\div (11-5)\)

Try it.

\(3\cdot 8+5\cdot 2\)

Solution

Try it.

\(4\cdot 7+3\cdot 5\)

Try it.

\(2+8(6+1)\)

Solution

Try it.

\(4+6(3+6)\)

Try it.

\(4\cdot 12/8\)

Solution

Try it.

\(2\cdot 36/6\)

Try it.

\(6+10/2+2\)

Solution

Try it.

\(9+12/3+4\)

Try it.

\((6+10)\div (2+2)\)

Solution

Try it.

\((9+12)\div (3+4)\)

Try it.

\(20\div 4+6\cdot 5\)

Solution

Try it.

\(33\div 3+8\cdot 2\)

Try it.

\(20\div (4+6)\cdot 5\)

Solution

Try it.

\(33\div (3+8)\cdot 2\)

Try it.

\({4}^{2}+{5}^{2}\)

Solution

Try it.

\({3}^{2}+{7}^{2}\)

Try it.

\({(4+5)}^{2}\)

Solution

Try it.

\({(3+7)}^{2}\)

Try it.

\(3(1+9\cdot 6)-{4}^{2}\)

Solution

149

Try it.

\(5(2+8\cdot 4)-{7}^{2}\)

Try it.

\(2[1+3(10-2)]\)

Solution

Try it.

\(5[2+4(3-2)]\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

\(\text{Add:}\ 43+69.\)
If you missed this problem, review .

Kusonyeza yankho

\(112\)
\(\text{Multiply:}\ (896)201.\)
If you missed this problem, review .

Kusonyeza yankho

\(180,096\)
\(\text{Divide:}\ 7,263\div 9.\)
If you missed this problem, review .

Kusonyeza yankho

\(807\)
Translate from algebra to words:
1. ⓐ \(\ 12+14\\)
2. ⓑ \(\ (30)(5)\\)
3. ⓒ \(\ 64\div 8\\)
4. ⓓ \(\ x-y\)
Kusonyeza yankho

ⓐ
\(12+14\)
12 plus 14
the sum of twelve and fourteen
ⓑ
\((30)(5)\)
30 times 5
the product of thirty and five
ⓒ
\(64\div 8\)
64 divided by 8
the quotient of sixty-four and eight
ⓓ
\(x-y\)
\(x\) minus \(y\)
the difference of \(x\) and \(y\)
Translate from algebra to words.
1. ⓐ \(\ 18+11\\)
2. ⓑ \(\ (27)(9)\\)
3. ⓒ \(\ 84\div 7\)
4. ⓓ \(p-q\)
Kusonyeza yankho
1. ⓐ 18 plus 11; the sum of eighteen and eleven
2. ⓑ 27 times 9; the product of twenty-seven and nine
3. ⓒ 84 divided by 7; the quotient of eighty-four and seven
4. ⓓ p minus q; the difference of p and q
Translate from algebra to words.
1. ⓐ \(\ 47-19\\)
2. ⓑ \(\ 72\div 9\\)
3. ⓒ \(\ m+n\\)
4. ⓓ \(\ (13)(7)\)
Kusonyeza yankho
1. ⓐ 47 minus 19; the difference of forty-seven and nineteen
2. ⓑ 72 divided by 9; the quotient of seventy-two and nine
3. ⓒ m plus n; the sum of m and n
4. ⓓ 13 times 7; the product of thirteen and seven
Translate from algebra to words:
1. ⓐ \(\ 20\le 35\)
2. ⓑ \(\ 11\ne 15-3\)
3. ⓒ \(\ 9>10\div 2\)
4. ⓓ \(\ x+2<10\)
Kusonyeza yankho

ⓐ
\(20\le 35\)
20 is less than or equal to 35
ⓑ
\(11\ne 15-3\)
11 is not equal to 15 minus 3
ⓒ
\(9>10\div 2\)
9 is greater than 10 divided by 2
ⓓ
\(x+2<10\)
\(x\) plus 2 is less than 10
Translate from algebra to words.
1. ⓐ \(\ 14\le 27\\)
2. ⓑ \(\ 19-2\ne 8\\)
3. ⓒ \(\ 12>4\div 2\\)
4. ⓓ \(\ x-7<1\)
Kusonyeza yankho
1. ⓐ fourteen is less than or equal to twenty-seven
2. ⓑ nineteen minus two is not equal to eight
3. ⓒ twelve is greater than four divided by two
4. ⓓ x minus seven is less than one
Translate from algebra to words.
1. ⓐ \(\ 19\ge 15\\)
2. ⓑ \(\ 7=12-5\\)
3. ⓒ \(\ 15\div 3<8\\)
4. ⓓ \(\ y-3>6\)
Kusonyeza yankho
1. ⓐ nineteen is greater than or equal to fifteen
2. ⓑ seven is equal to twelve minus five
3. ⓒ fifteen divided by three is less than eight
4. ⓓ y minus three is greater than six
The information in compares the fuel economy in miles-per-gallon (mpg) of several cars. Write the appropriate symbol \(\text{=},\text{<},\text{or}\ \text{>}\) in each expression to compare the fuel economy of the cars.
1. ⓐ MPG of Prius_____ MPG of Mini Cooper
2. ⓑ MPG of Versa_____ MPG of Fit
3. ⓒ MPG of Mini Cooper_____ MPG of Fit
4. ⓓ MPG of Corolla_____ MPG of Versa
5. ⓔ MPG of Corolla_____ MPG of Prius
Kusonyeza yankho

ⓐ
MPG of Prius____MPG of Mini Cooper
Find the values in the chart. 48____27
Compare. 48 > 27
MPG of Prius > MPG of Mini Cooper
ⓑ
MPG of Versa____MPG of Fit
Find the values in the chart. 26____27
Compare. 26 < 27
MPG of Versa < MPG of Fit
ⓒ
MPG of Mini Cooper____MPG of Fit
Find the values in the chart. 27____27
Compare. 27 = 27
MPG of Mini Cooper = MPG of Fit
ⓓ
MPG of Corolla____MPG of Versa
Find the values in the chart. 28____26
Compare. 28 > 26
MPG of Corolla > MPG of Versa
ⓔ
MPG of Corolla____MPG of Prius
Find the values in the chart. 28____48
Compare. 28 < 48
MPG of Corolla < MPG of Prius
Use to fill in the appropriate \(\text{symbol},\text{=},\text{<},\text{or}\ \text{>}.\)
1. ⓐ MPG of Prius_____MPG of Versa
2. ⓑ MPG of Mini Cooper_____ MPG of Corolla
Kusonyeza yankho
1. ⓐ >
2. ⓑ <
Use to fill in the appropriate \(\text{symbol},\text{=},\text{<},\text{or}\ \text{>}.\)
1. ⓐ MPG of Fit_____ MPG of Prius
2. ⓑ MPG of Corolla _____ MPG of Fit
Kusonyeza yankho
1. ⓐ <
2. ⓑ >

Determine if each is an expression or an equation:

ⓐ \(\ 16-6=10\)
ⓑ \(\ 4\cdot 2+1\)
ⓓ \(\ y+8=40\)

Kusonyeza yankho

ⓐ \(\ 16-6=10\)	This is an equation—two expressions are connected with an equal sign.
ⓑ \(\ 4\cdot 2+1\)	This is an expression—no equal sign.
ⓒ \(\ x\div 25\)	This is an expression—no equal sign.
ⓓ \(\ y+8=40\)	This is an equation—two expressions are connected with an equal sign.

Determine if each is an expression or an equation:
1. ⓐ \(\ 23+6=29\\)
2. ⓑ \(\ 7\cdot 3-7\)
Kusonyeza yankho
1. ⓐ equation
2. ⓑ expression
Determine if each is an expression or an equation:
1. \(\ y\div 14\\)
2. \(\ x-6=21\)
Kusonyeza yankho
1. ⓐ expression
2. ⓑ equation
Write each expression in exponential form:
1. ⓐ \(\ 16\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16\)
2. ⓑ \(\ 9\cdot 9\cdot 9\cdot 9\cdot 9\)
3. ⓒ \(\ x\cdot x\cdot x\cdot x\)
4. ⓓ \(\ a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\)
Kusonyeza yankho

ⓐ The base 16 is a factor 7 times. \({16}^{7}\)
ⓑ The base 9 is a factor 5 times. \({9}^{5}\)
ⓒ The base \(x\) is a factor 4 times. \({x}^{4}\)
ⓓ The base \(a\) is a factor 8 times. \({a}^{8}\)
Write each expression in exponential form:
\(41\cdot 41\cdot 41\cdot 41\cdot 41\)

Kusonyeza yankho

41⁵
Write each expression in exponential form:
\(7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7\cdot 7\)

Kusonyeza yankho

7⁹
Write each exponential expression in expanded form:
1. ⓐ \(\ {8}^{6}\\)
2. ⓑ \(\ {x}^{5}\)
Kusonyeza yankho

ⓐ The base is \(8\) and the exponent is \(6,\) so \({8}^{6}\) means \(8\cdot 8\cdot 8\cdot 8\cdot 8\cdot 8\)
ⓑ The base is \(x\) and the exponent is \(5,\) so \({x}^{5}\) means \(x\cdot x\cdot x\cdot x\cdot x\)
Write each exponential expression in expanded form:
1. ⓐ \(\ {4}^{8}\)
2. ⓑ \(\ {a}^{7}\)
Kusonyeza yankho
1. ⓐ 4 · 4 · 4 · 4 · 4 · 4 · 4 · 4
2. ⓑ a · a · a · a · a · a · a
Write each exponential expression in expanded form:
1. ⓐ \({\ 8}^{8}\\)
2. ⓑ \(\ {b}^{6}\)
Kusonyeza yankho
1. ⓐ 8 · 8 · 8 · 8 · 8 · 8 · 8 · 8
2. ⓑ b · b · b · b · b · b
Simplify: \({3}^{4}.\)

Kusonyeza yankho

\({3}^{4}\)
Expand the expression. \(3⋅3⋅3⋅3\)
Multiply left to right. \(9⋅3⋅3\)
\(27⋅3\)
Multiply. \(81\)
Simplify:
1. ⓐ \(\ {5}^{3}\\)
2. ⓑ \(\ {1}^{7}\)
Kusonyeza yankho
1. ⓐ 125
2. ⓑ 1
Simplify:
1. ⓐ \(\ {7}^{2}\\)
2. ⓑ \(\ {0}^{5}\)
Kusonyeza yankho
1. ⓐ 49
2. ⓑ 0
Simplify the expressions:
1. ⓐ \(\ 4+3\cdot 7\\)
2. ⓑ \(\ (4+3)\cdot 7\)
Kusonyeza yankho

ⓐ
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply first.
Add.
ⓑ
Are there any parentheses? Yes.
Simplify inside the parentheses.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply.
Simplify the expressions:
1. ⓐ \(\ 12-5\cdot 2\\)
2. ⓑ \(\ (12-5)\cdot 2\)
Kusonyeza yankho
1. ⓐ 2
2. ⓑ 14
Simplify the expressions:
1. ⓐ \(\ 8+3\cdot 9\\)
2. ⓑ \((8+3)\cdot 9\)
Kusonyeza yankho
1. ⓐ 35
2. ⓑ 99
Simplify:
1. ⓐ \(\ 18\div 9\cdot 2\\)
2. ⓑ \(\ 18\cdot 9\div 2\)
Kusonyeza yankho

ⓐ
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply and divide from left to right. Divide.
Multiply.
ⓑ
Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply and divide from left to right.
Multiply.
Divide.
Simplify:
\(42\div 7\cdot 3\)

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18
Simplify:
\(12\cdot 3\div 4\)

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9

Simplify: \(18\div 6+4(5-2).\)

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Parentheses? Yes, subtract first.
Exponents? No.
Multiplication or division? Yes.
Divide first because we multiply and divide left to right.
Any other multiplication or division? Yes.
Multiply.
Any other multiplication or division? No.
Any addition or subtraction? Yes.

Simplify:
\(30\div 5+10(3-2)\)

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16
Simplify:
\(70\div 10+4(6-2)\)

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23

\(\text{Simplify:}\ 5+{2}^{3}+3[6-3(4-2)].\)

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Are there any parentheses (or other grouping symbol)? Yes.
Focus on the parentheses that are inside the brackets.
Subtract.
Continue inside the brackets and multiply.
Continue inside the brackets and subtract.
The expression inside the brackets requires no further simplification.
Are there any exponents? Yes.
Simplify exponents.
Is there any multiplication or division? Yes.
Multiply.
Is there any addition or subtraction? Yes.
Add.
Add.

Simplify:
\(9+{5}^{3}-[4(9+3)]\)

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86
Simplify:
\({7}^{2}-2[4(5+1)]\)

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1
Simplify: \({2}^{3}+{3}^{4}\div 3-{5}^{2}.\)

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If an expression has several exponents, they may be simplified in the same step.
Simplify exponents.
Divide.
Add.
Subtract.
Simplify:
\({3}^{2}+{2}^{4}\div 2+{4}^{3}\)

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81
Simplify:
\({6}^{2}-{5}^{3}\div 5+{8}^{2}\)

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75
\(5\cdot 6\)

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5 times 6, the product of five and six

Symbols used here

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\approx

approximately equal
Equal to the precision shown, not exactly.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Use the Language of Algebra

Use variables and algebraic symbols
Identify expressions and equations
Simplify expressions with exponents
Simplify expressions using the order of operations
The
The
The
The

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Sankhani wanu

Parts of this page are adapted from OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Zambiri pa Arithmetic

Order of operations Fractions Percentages Powers and roots

ⓐ
	MPG of Prius____MPG of Mini Cooper
Find the values in the chart.	48____27
Compare.	48 > 27
	MPG of Prius > MPG of Mini Cooper

ⓑ
	MPG of Versa____MPG of Fit
Find the values in the chart.	26____27
Compare.	26 < 27
	MPG of Versa < MPG of Fit

ⓒ
	MPG of Mini Cooper____MPG of Fit
Find the values in the chart.	27____27
Compare.	27 = 27
	MPG of Mini Cooper = MPG of Fit

ⓓ
	MPG of Corolla____MPG of Versa
Find the values in the chart.	28____26
Compare.	28 > 26
	MPG of Corolla > MPG of Versa

ⓔ
	MPG of Corolla____MPG of Prius
Find the values in the chart.	28____48
Compare.	28 < 48
	MPG of Corolla < MPG of Prius

	\({3}^{4}\)
Expand the expression.	\(3⋅3⋅3⋅3\)
Multiply left to right.	\(9⋅3⋅3\)
	\(27⋅3\)
Multiply.	\(81\)

ⓐ

Are there any parentheses? No.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply first.
Add.

ⓑ

Are there any parentheses? Yes.
Simplify inside the parentheses.
Are there any exponents? No.
Is there any multiplication or division? Yes.
Multiply.


If an expression has several exponents, they may be simplified in the same step.
Simplify exponents.
Divide.
Add.
Subtract.