### 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]