Idiot\'s Guides Basic Math and Pre-Algebra

104 Part 2: Into the Unknown

CHECK POINT

Write each sentence using variables.

1. The quotient of some number x and twelve is greater than four.


2. The difference of a number t and nineteen is twenty-two.


3. The sum of a number n and the quantity n increased by three is one hundred
   fifty-four.


4. The product of a number y and the quantity five less than y is eighty-four.


5. The quotient of a number p and the quantity p increased by one is two.

Multiplying with Variables .................................................................................................

If variables take the place of numbers, then it makes sense that you should be able to do arithmetic

with variables, because you do arithmetic with numbers. Not knowing what number the variable

stands for is an obvious problem when it comes to arithmetic, but there are some arithmetic opera-

tions you can do with variables, if you work carefully.

Let’s start with multiplication, because that’s probably the most common operation and the one

with the fewest restrictions or dangers. If you need to multiply a constant, like 4, by a variable,

like t, you can do that. Because you don’t know what number t stands for, you can’t give a number

as the answer, but you can write 4t.

If you needed to multiply a constant, like -7, by the product of a constant and a variable, like 4t,

you apply the associative property. -7™(4t) = (-7™4)™t = -28t. You multiply the constant by the

coefficient and keep the variable as the final factor.

Things get interesting when a multiplication involves more than one variable. If you wanted

to multiply -7t by 4t, you could call on the associative property and the commutative property.

-7t™ 4 t = (-7™t)™(4™t) = (-7™4)™(t™t) = -28™t™t. Do you remember the shortcut for writing repeated

multiplication, like t™t? You can use an exponent. -28™t™t = -28t^2.

Working with Exponents

Exponents, you remember, are symbols for repeated multiplication. The expression 5^3 , for

example, means 5·5·5. That ability to express repeated multiplication with exponents is even more

important when you’re working with variables, because you don’t know what number the variable

stands for, so you can’t evaluate the multiplication. When you write bn you say that you want to

use b as a factor n times. The expression b^5 means b™b™b™b™b, but until you know what number b

stands for, that’s all you can say.