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Обучение Руководства > Prealgebra

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
3+53+5 3 plus 53\text{ plus }5 the sum of three and five
n−1n - 1 nn minus one the difference of nn and one
6⋅76\cdot 7 6 times 76\text{ times }7 the product of six and seven
xy\frac{x}{y} xx divided by yy the quotient of xx and yy
  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=83+5=8 The sum of three and five is equal to eight.
n−1=14n - 1=14 nn minus one equals fourteen.
6⋅7=426\cdot 7=42 The product of six and seven is equal to forty-two.
x=53x=53 xx is equal to fifty-three.
y+9=2y−3y+9=2y - 3 yy plus nine is equal to two yy 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:
  1. 16−6=1016 - 6=10
  2. 4⋅2+14\cdot 2+1
  3. x÷25x\div 25
  4. y+8=40y+8=40
Solution
1. 16−6=1016 - 6=10 This is an equation—two expressions are connected with an equal sign.
2. 4⋅2+14\cdot 2+1 This is an expression—no equal sign.
3. x÷25x\div 25 This is an expression—no equal sign.
4. y+8=40y+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⋅2+14\cdot 2+1 we’d first multiply 4⋅24\cdot 2 to get 88 and then add the 11 to get 99. 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⋅2+14\cdot 2+1 8+18+1 99

Suppose we have the expression 2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅22\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⋅2⋅22\cdot 2\cdot 2 as 23{2}^{3} and 2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅22\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2 as 29{2}^{9}. In expressions such as 23{2}^{3}, the 22 is called the base and the 33 is called the exponent. The exponent tells us how many factors of the base we have to multiply.

The image shows the number two with the number three, in superscript, to the right of the two. The number two is labeled as means multiply three factors of 2\text{means multiply three factors of 2} We say 23{2}^{3} is in exponential notation and 2⋅2⋅22\cdot 2\cdot 2 is in expanded notation.

Exponential Notation

For any expression an,a{a}^{n},a is a factor multiplied by itself nn times if nn is a positive integer. an means multiply n factors of a{a}^{n}\text{ means multiply }n\text{ factors of }a At the top of the image is the letter a with the letter n, in superscript, to the right of the a. The letter a is labeled as The expression an{a}^{n} is read aa to the nth{n}^{th} power.
For powers of n=2n=2 and n=3n=3, we have special names.

a2a^2 is read as "aa squared"

a3a^3 is read as "aa cubed"

  The table below lists some examples of expressions written in exponential notation.
Exponential Notation In Words
72{7}^{2} 77 to the second power, or 77 squared
53{5}^{3} 55 to the third power, or 55 cubed
94{9}^{4} 99 to the fourth power
125{12}^{5} 1212 to the fifth power
 

example

Write each expression in exponential form:
  1. 16⋅16⋅16⋅16⋅16⋅16⋅1616\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16\cdot 16
  2. 9⋅9⋅9⋅9⋅9\text{9}\cdot \text{9}\cdot \text{9}\cdot \text{9}\cdot \text{9}
  3. x⋅x⋅x⋅xx\cdot x\cdot x\cdot x
  4. a⋅a⋅a⋅a⋅a⋅a⋅a⋅aa\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a\cdot a

Answer: Solution

1. The base 1616 is a factor 77 times. 167{16}^{7}
2. The base 99 is a factor 55 times. 95{9}^{5}
3. The base xx is a factor 44 times. x4{x}^{4}
4. The base aa is a factor 88 times. a8{a}^{8}

  In the video below we show more examples of how to write an expression of repeated multiplication in exponential form. https://youtu.be/HkPGTmAmg_s

example

Write each exponential expression in expanded form:
  1. 86{8}^{6}
  2. x5{x}^{5}

Answer: Solution 1. The base is 88 and the exponent is 66, so 86{8}^{6} means 8⋅8⋅8⋅8⋅8⋅88\cdot 8\cdot 8\cdot 8\cdot 8\cdot 8 2. The base is xx and the exponent is 55, so x5{x}^{5} means x⋅x⋅x⋅x⋅xx\cdot x\cdot x\cdot x\cdot x

    To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

example

Simplify: 34{3}^{4}

Answer: Solution

34{3}^{4}
Expand the expression. 3⋅3⋅3⋅33\cdot 3\cdot 3\cdot 3
Multiply left to right. 9⋅3⋅39\cdot 3\cdot 3
27⋅327\cdot 3
Multiply. 8181

   

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