Learning Outcomes
- Identify and write mathematical expressions using symbols and words
- Identify and write mathematical equations using symbols and words
- Identify the difference between an expression and an equation
- Use exponential notation to express repeated multiplication
- Write an exponential expression in expanded form
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 |
|---|---|---|
| [latex]3+5[/latex] | [latex]3\text{ plus }5[/latex] | the sum of three and five |
| [latex]n - 1[/latex] | [latex]n[/latex] minus one | the difference of [latex]n[/latex] and one |
| [latex]6\cdot 7[/latex] | [latex]6\text{ times }7[/latex] | the product of six and seven |
| [latex]\frac{x}{y}[/latex] | [latex]x[/latex] divided by [latex]y[/latex] | the quotient of [latex]x[/latex] and [latex]y[/latex] |
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 |
|---|---|
| [latex]3+5=8[/latex] | The sum of three and five is equal to eight. |
| [latex]n - 1=14[/latex] | [latex]n[/latex] minus one equals fourteen. |
| [latex]6\cdot 7=42[/latex] | The product of six and seven is equal to forty-two. |
| [latex]x=53[/latex] | [latex]x[/latex] is equal to fifty-three. |
| [latex]y+9=2y - 3[/latex] | [latex]y[/latex] plus nine is equal to two [latex]y[/latex] minus three. |
Expressions and Equations
An expression is a number, a variable, or a combination of numbers and variables and operation symbols.
An equation is made up of two expressions connected by an equal sign.
example
Determine if each is an expression or an equation:
- [latex]16 - 6=10[/latex]
- [latex]4\cdot 2+1[/latex]
- [latex]x\div 25[/latex]
- [latex]y+8=40[/latex]
Solution
| 1. [latex]16 - 6=10[/latex] | This is an equation—two expressions are connected with an equal sign. |
| 2. [latex]4\cdot 2+1[/latex] | This is an expression—no equal sign. |
| 3. [latex]x\div 25[/latex] | This is an expression—no equal sign. |
| 4. [latex]y+8=40[/latex] | This is an equation—two expressions are connected with an equal sign. |
try it
Simplify Expressions with Exponents
You have simplified many expressions so far using the four main mathematical operations. To simplify a numerical expression means to do all the math possible. For example, to simplify [latex]4\cdot 2+1[/latex] we’d first multiply [latex]4\cdot 2[/latex] to get [latex]8[/latex] and then add the [latex]1[/latex] to get [latex]9[/latex]. 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:
[latex]4\cdot 2+1[/latex]
[latex]8+1[/latex]
[latex]9[/latex]
However, there are other mathematical notations used to simplify the numbers we are working with. Suppose we have the expression [latex]2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2[/latex]. 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 [latex]2\cdot 2\cdot 2[/latex] as [latex]{2}^{3}[/latex] and [latex]2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2[/latex] as [latex]{2}^{9}[/latex]. In expressions such as [latex]{2}^{3}[/latex], the [latex]2[/latex] is called the base and the [latex]3[/latex] is called the exponent. The exponent tells us how many factors of the base we have to multiply.
[latex]\text{means multiply three factors of 2}[/latex]
We say [latex]{2}^{3}[/latex] is in exponential notation and [latex]2\cdot 2\cdot 2[/latex] is in expanded notation.
Exponential Notation
For any expression [latex]{a}^{n},a[/latex] is a factor multiplied by itself [latex]n[/latex] times if [latex]n[/latex] is a positive integer.
[latex]{a}^{n}\text{ means multiply }n\text{ factors of }a[/latex]
The expression [latex]{a}^{n}[/latex] is read [latex]a[/latex] to the [latex]{n}^{th}[/latex] power.
For powers of [latex]n=2[/latex] and [latex]n=3[/latex], we have special names.
[latex]a^2[/latex] is read as “[latex]a[/latex] squared”
[latex]a^3[/latex] is read as “[latex]a[/latex] cubed”
The table below lists some examples of expressions written in exponential notation.
| Exponential Notation | In Words |
|---|---|
| [latex]{7}^{2}[/latex] | [latex]7[/latex] to the second power, or [latex]7[/latex] squared |
| [latex]{5}^{3}[/latex] | [latex]5[/latex] to the third power, or [latex]5[/latex] cubed |
| [latex]{9}^{4}[/latex] | [latex]9[/latex] to the fourth power |
| [latex]{12}^{5}[/latex] | [latex]12[/latex] to the fifth power |
example
Write each expression in exponential form:
- [latex]16\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16[/latex]
- [latex]\text{9}\cdot \text{9}\cdot \text{9}\cdot \text{9}\cdot \text{9}[/latex]
- [latex]x\cdot x\cdot x\cdot x[/latex]
- [latex]a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a[/latex]
try it
In the video below we show more examples of how to write an expression of repeated multiplication in exponential form.
example
Write each exponential expression in expanded form:
- [latex]{8}^{6}[/latex]
- [latex]{x}^{5}[/latex]
try it
To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.
example
Simplify: [latex]{3}^{4}[/latex]
try it
