Identifying Expressions and Equations
Learning Outcomes
- Identify and write mathematical expressions using words and symbols
- Identify and write mathematical equations using words and symbols
- 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] |
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]
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. |
Simplify Expressions with Exponents
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-display]8+1[/latex-display] [latex]9[/latex]
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-display]\text{means multiply three factors of 2}[/latex-display] 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-display]{a}^{n}\text{ means multiply }n\text{ factors of }a[/latex-display] The expression [latex]{a}^{n}[/latex] is read [latex]a[/latex] to the [latex]{n}^{th}[/latex] power.[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]
Answer: Solution
1. The base [latex]16[/latex] is a factor [latex]7[/latex] times. | [latex]{16}^{7}[/latex] |
2. The base [latex]9[/latex] is a factor [latex]5[/latex] times. | [latex]{9}^{5}[/latex] |
3. The base [latex]x[/latex] is a factor [latex]4[/latex] times. | [latex]{x}^{4}[/latex] |
4. The base [latex]a[/latex] is a factor [latex]8[/latex] times. | [latex]{a}^{8}[/latex] |
example
Write each exponential expression in expanded form:- [latex]{8}^{6}[/latex]
- [latex]{x}^{5}[/latex]
Answer: Solution 1. The base is [latex]8[/latex] and the exponent is [latex]6[/latex], so [latex]{8}^{6}[/latex] means [latex]8\cdot 8\cdot 8\cdot 8\cdot 8\cdot 8[/latex] 2. The base is [latex]x[/latex] and the exponent is [latex]5[/latex], so [latex]{x}^{5}[/latex] means [latex]x\cdot x\cdot x\cdot x\cdot x[/latex]
example
Simplify: [latex]{3}^{4}[/latex]Answer: Solution
[latex]{3}^{4}[/latex] | |
Expand the expression. | [latex]3\cdot 3\cdot 3\cdot 3[/latex] |
Multiply left to right. | [latex]9\cdot 3\cdot 3[/latex] |
[latex]27\cdot 3[/latex] | |
Multiply. | [latex]81[/latex] |
Licenses & Attributions
CC licensed content, Shared previously
- Example: Write Repeated Multiplication in Exponential Form. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Question ID: 144735, 144737, 144744, 144745. Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].