Real life mathematics require us to be able to take a problem and write it in a form from which we can arrive at an answer. To do this effectively, we need to know what the problem is saying. Some initial vocabulary will start us along that path.

Variable - a letter that can represent any number. You may use any letter except o or i. An o looks to much like a zero. An i has a special meaning in higher mathematics.

Algebraic Expression - an expression that contains at least one variable.

Examples:
3x - 5
ab + c
^w/₅

Numerical Expression - an expression that contains only numbers.

Examples:
3(4) - 5
5(2) + 7
⁸/₅

In writing either Algebraic or Numerical expressions, we need to know the words that indicate the basic mathematical operations.

Addition	the sum of Increased by plus more than added to the total of	Special note:  more than tells us to add that number, so five more than a number tells us to write x + 5
Subtraction	the difference of decreased by minus less than subtract from	Special note:  less than tells us to subtract that number, so five less than a number tells us to write x - 5

Notice - in the last two examples, you represent a number with a variable. I chose to use x, but you can use any letter except o or i.

Multiplication	the product of multiplied by times	The numbers in a multiplication problem are called factors. The answer is the product.
Division	the quotient of divided by the ratio of	w/5 means w ÷ 5 A ratio is a fraction, so a ratio of 4 to 5 is written as 4/5 which is a division problem
Powers	xa where	x is called the base a is called the exponent xa is a power

An exponent tells us how many equal factors there are.

Since factors are numbers in a multiplication problem, the exponent tells us how many times to multiply the base times itself.

Examples:
3⁴ = 3·3·3·3
a·a·a·a·a = a⁵
X³ = x·x·x
m·m·n·n·n = m²n³

Special names for special powers:

x² can be read x to the 2^nd power or x squared

x³ can be read x to the 3^rd power or x cubed

Examples: Change the following verbal expressions into mathematical expressions.

1.  A number increased by five

Reasoning:

a.  a number - we are not told which number, so it could be any number and is represented by a variable
b.  increased by tells us to add
c.  increased by five tells us to add five

Answer:  x + 5

2.  The ratio of seven and a number

Reasoning:

a.  ratio tells us to divide
b.  a number - represented by a variable

Answer:  ⁷/_y

3.  Eight less than three times a number

Reasoning:

a.  less than tell us to subtract eight
b.  three times a number - multiply three times a variable

Answer:  3a - 8

4.  The product of a number and ten

Reasoning:

a.  product tells us to multiply

Answer:  10n  Note: when multiplying, we will write the number first.

Examples: change the following Algebraic expressions to Verbal Expressions:

1.  4m - 5

Reasoning:  Look at your table for subtraction and choose any of the five possible translations. You need to write just one of the possibilities.

Four times a number decreased by five

The difference of four times a number and five

Five less than four times a number

Five subtracted from four times a number

Four times a number minus five

2.   w
    ¾¾
      2³

Reasoning:

a.  w is a variable so it can be any number, so we write a number

b.  the fraction ^w/₂ indicates division, so choose one of the division translations

c.  2³ can be read as two cubed or two to the 3^rd power; either one is acceptable

A number divided by two cubed
The quotient of a number and two to the 3^rd power
The ratio of a number and two cubed

As you can see from the previous two examples, you can choose any of the possible translations and your answer will be acceptable. You must be familiar with all the possibilities.

Examples: Change the following from expenential form to factored form:

1.	6a2 = 6·a·a	The exponent of two goes only with the a
2.	(6a)2 = 6a·6a	The ( ) indicate that everything inside the ( ) is squared
3.	m3 n2 = m·m·m·n·n
4.	6 a b4 = 6·a·b·b·b·b	Note: neither the six nor the a has an exponent. The exponents are understood to be one. We do not write exponents of one.

Examples: Change from factored form to exponential form:

1.	w·w·w·w·w = w5
2.	3a2 · 3a2 · 3a2 = (3a2)3	Note: There are three equal factors of 3a2. To indicate that the whole thing is cubed, we must use ( ).

As indicated previously, we can read x² as x squared and x³ as x cubed. We can relate this directly into the Geometry formulas for the area of a square and the volume of a cube.

Area of a square:  A = s² where s is the length of one side. Remember that the length of the sides in a square and a cube are of equal length!

Volume of a cube:  V = s³