Basic Math and Pre-Algebra
PART 2. Into the Unknown
CHAPTER 8. Variables and Expressions
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 operations 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∙4t = (-7∙t)∙(4∙t) = (-7∙4)∙(t∙1) = -28∙t∙t. Do you remember the shortcut for writing repeated multiplication, like t∙t? You can use an exponent. -28∙t∙t = -28t2.
Working with Exponents
Exponents, you remember, are symbols for repeated multiplication. The expression 53, 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 b5 means b∙b∙b∙b∙b, but until you know what number b stands for, that’s all you can say.
When you work with exponents, there are three basic rules to remember.
Multiplication: When you multiply powers of the same base, keep the base and add the exponents. This means that x2∙x3 = x5 because x2∙x3 = (x∙x)∙(x∙x∙x) = x5. Don’t forget that when you have a single variable, like y, even though you don’t see an exponent, y is y1. Multiplying y∙y3 gives you y4. Multiplying a power of a variable by the variable again raises the exponent.
Division: When you divide powers of the same base, keep the base and subtract the exponents. This means that Don’t forget that a variable to the zero power equals 1.
Powers: When you raise a power to a power, keep the base and multiply the exponents. This one doesn’t come up as often, but it means that (t7)3 = t21. Remember that exponents work only on what they touch, so 3x2 means 3∙x∙x, because the exponent is only touching the x, but (3x)2 = (3x)∙(3x) because the exponent is touching the parenthesis, and so the whole quantity is squared.
Ways to Multiply
You can multiply a constant by a variable or a power of a variable: -5∙x2 = -5x2.
You can multiply a constant by a product of a constant and a variable: 13∙4t = 52t.
You can multiply a variable or a power of a variable by a variable or a power of a variable: y∙y3 = y4.
You can multiply a variable or a power of a variable by a product of a constant and a variable or a power of a variable.: a2∙6a3 = 6a5.
You can multiply the product of a constant and a variable (or a power of a variable) by a product of a constant and a variable (or a power of a variable): -8x2∙4x2 = -32x4.
And you can even multiply a constant, a variable, or the product of a constant and a variable by an expression: 2x(3x - 5) = 2x(3x) - 2x(5) = 6x2 - 10x and 3t2(t2 + 2t + 1) = 3t2(t2) + 3t2(2t) + 3t2(1) = 3t4 + 6t3 + 3t2.
In case you were wondering, you can multiply with two different variables. You just simplify as much as possible: 5x(2y) = 10xy and 2ab2(3a2b) = 2∙3∙a∙a2∙b2∙b = 6a3b3.
Put each product in simplest form.
10. 7z(3z + 5)