Variables and Expressions

***Attention Students***
Use the form at the bottom of the page to submit your work.

Hello and welcome to Algebra!! I hope you will feel free to ask any questions in the classroom, or email me anytime. I check in both places regularly and I am here to help you!!

Let's begin with Variables and Expressions.

In algebra, variables are symbols that are used to represent unspecified numbers. Any letter may be used as a variable. The most commonly used are x and y, but any letter can be used- p, q, a, b, z, etc. They are usually represented with lower case letters, not capital letters. We reserve the capital letters to represent formulas.

An algebraic expression consists of one or more numbers and variables along with one or more arithmetic operations. Here are some examples of algebraic expressions:

b + 4     is the same as   "b plus 4"
s/t - 1    is the same as   "s divided by t minus 1"
a * 4n is the same as   "a times 4 times n"
8rs/3k is the same as   "8 times the quantity (r times s) divided by the
                                              quantity of( 3 times k)"

In a multiplication expression, the quantities being multiplied are called factors and the result is called the product.

Example:

4 * 5 * 8 = 160

4, 5, and 8 are called factors and 160 is called product

In algebraic expressions, a raised dot, (I will use the * symbol), which should be located above the 8 on your keyboard, or parentheses are often used to indicate multiplication. When variables are used to represent factors, the multiplication sign is usually omitted. Here are a few ways to represent the product of a and b:

a * b

a(b)

(a)b

(a)(b)

Fraction bars are used to indicate division. Here are a several examples:

x/3 "x divided by 3"

(x + 4)/ y "the quantity of x +4 divided by y"

This means that you divide the numerator, top (x + 4) by the denominator (bottom)

To solve some verbal problems in mathematics, you can create an algebraic expression by translating words into mathematical symbols. The charts below shows some of the words that are used to indicate mathematical operations. These charts are useful tools. Either print them out or write them down and keep them handy for reference .

ADDITION

+ the sum of

+ increased by

+ plus

+ more than

+ added to

+ the total of

SUBTRACTION

- the difference of

- decreased by

- minus

- less than

- subtracted from

MULTIPLICATION

* the product of

* multiplied by

* times

* the word "of"

DIVISION

/ the quotient of

/ divided by

/ the ratio of

LET'S TRY SOME EXAMPLES TOGETHER:

Example 1

Write an algebraic expression for the sum of a and b.

" Sum " means addition so we add a and b together to get the algebraic expression a+b

Example 2

Write an algebraic expression for a number k divided by a number n.

Of course,"divided by" means division, so we would get the algebraic expression is k/n

Let's do a few of these examples the other way now.
Example 3

Write a verbal expression for 4x.

Since there is no operation symbol between 4 and x, multiplication is ALWAYS assumed, the verbal expression is the product of 4 and x, or 4 times x.

Example 4

Write a verbal expression for z- 6.

A verbal expression is six less than z, or the difference between z and 6.

In an expression like 10^3, 10 is the base and 3 is the exponent. The exponent indicates the number of times the base is used as a factor. With paper and pencil, we just place a little 3 right next to the right hand corner of the 10. But, to use the exponent on the computer, the character located above the number 6 should be used. We also call this character above the 6 a "carrot". So...

10^3 means 10 raised to the third power, or 10* 10 * 10, which equals 1000

Example 5

a^2 means raised to the second power, or a * a

Example 6

8^1 means 8 to the first power or,8

Example 7

8^2 means 8 to the second power,or 8 squared, or 8 *8 = 64

Example 8

8^3 means 8 to the third power,or 8 cubed, or 8 * 8* 8 = 512

Example 9

8^4 means 8 to the fourth power,or 8 * 8 * 8 * 8 = 4096

Example 10

6n^5 means 6 times n to the fifth power, or 6 * n * n * n * n * n

Try this example:

Example 11

Evaluate 2^5.

2^5 = 2 *2 * 2 * 2 * 2 = 32

Example 12

Evaluate 3^3.

3^3 = 3 *3 * 3 = 27

Example 13

Write an algebraic expression for the expression the cube of k increased by seven. The cube of a number means an exponent, so the cube of k means k raised to the third power then add 7.
The result would be an algebraic expression is k^3 + 7

We will learn how to evaluate these much later in Algebra.

Algebra 1 - Lesson 1

Take the test below. If you get 3 or fewer wrong, it will be assumed that you understand the lesson, and you can go on immediately to the next lesson.

If you get more than 3 wrong, you will be asked to resubmit the wrong answers and show your work. The teacher will then look at your work and give you advice on what you are doing wrong.

Name:
Enter your correct email address:

Write an algebraic expression for each verbal expression. These are similar to examples 1 and 2.

1. a number increased by 17

	  x + 17       
	  x - 17       
	  x / 17       
	  x * 7       
	  x = 17       
	  x > 17


2.  twice the cube of a number

	  6 * y       
	  2 * y^3       
	  3 * y^2       
	  (2 * y)^3       
	  2 * y^4       
	  5 * y


3.  one-half the square of a number

	  1 * 2^z       
	  z / 2^3       
	  (1/2)^z       
	  3 * z^(1/2)       
	  1 /(z^ * 3)       
	  1/2 * z^3




4.  94 increased by twice a number

	  94 * 2a       
	  94^2a       
	  94/2 * a       
	  94 + 2 * a       
	  2 - 94a       
	  94 - 2a


5.  seven times a number

	  7b       
	  7/b       
	  7^b       
	  b^7       
	  7 - b       
	  b/7


6.  a number to the sixth power

	  c * 6       
	  c/(c^6)       
	  c^6       
	  6/c       
	  c + 6       
	  c + 5c


7.  six times a number decreased by 17

	  6(d - 17)       
	  6d - 17       
	  6 - 17d       
	  (6 - 17) * d       
	  11d       
	  6^17 - d


8.  three fourths of the square of a number

	  3/4 * e^2       
	  3/4 * 2e       
	  3 * 4 / e^2       
	  3^4 * 2 * e       
	  3/4 * e/2       
	  3 * e/4^2


 





    Write a verbal expression for each algebraic expression. These are similar to examples 3 and 4.




9.  m - 1 

    


	  one plus a number       
	  a number raised to the first power       
	  a number divided by one       
	  one less than a number       
	  a number less than one       
	  one divided by a number


10.  n^4 

	  four times a number       
	  four less than a number       
	  four divided by a number       
	  a number raised to the fourth power       
	  four divided by a number       
	  four minus a number


11.  8y^2 

	  eight divided by a number squared       
	  eight times a number divided by two       
	  half of eight to the power of a number       
	  eight times two more than a number       
	  eight times the square of a number       
	  eight squared times a number


12.  xy

	  one number to the power of another number       
	  one number divided by another number       
	  one number plus another number       
	  one number times another number       
	  one number times the square of another number       
	  two numbers equal to each other


13.  5^3

	  five to the power of another number       
	  the cube of fifteen       
	  five divided by three       
	  five times three divided by another number       
	  five cubed       
	  three to the power of five


14.  z^7+2

	  two more than the seventh power of a number       
	  a number divided by seven plus two       
	  one-half times a number times seven       
	  one-seventh of a number squared       
	  seven squared times a number       
	  seven times a number plus two


 





    Use your calculator to evaluate each expression. 




15.  5^3

    


	  15       
	  53       
	  500       
	  27       
	  5000       
	  125


16.  3^6

	  36       
	  18       
	  1/2       
	  729       
	  27       
	  9


17.  (6.2)^6

	  37.2       
	  1.033       
	  0.968       
	  626       
	  56800.236       
	  10.333


18.  5^5 + 6^6 + 7^7

	  873,324       
	  110       
	  198       
	  3       
	  22       
	  10,048


19.  2^2 * 3^3 * 4^4

	  99       
	  27,648       
	  31,944       
	  22       
	  3       
	  16.5


20.  (4.8)^5

	  24       
	  0.96       
	  9.6       
	  240       
	  131,072       
	  2548.040



21.  The White House has 132 rooms, including a barber shop and a bowling alley. The main building is shaped like a rectangle that is 168 feet long and 85 feet wide. What is the area of the main building of the White House?

	  253 ft       
	  1.4^189       
	  14,280 sq ft       
	  1.976 sq ft       
	  197.6 ft       
	  83 ft









Content © Compuhigh.com