Multiply Variable Expressions

We all know how to multiply, but sometimes multiplying expressions that contain variables can get confusing.  The main point you want to remember is that multiplication works the same, no matter what you are multiplying:

1. There is a commutatative property of multiplication.  That means that you can multiply numbers in any order.
2. When you multiply a number or a variable times itself, you capture that with an exponent (so $3\times 3 = 3^2$ and $x \times x = x^2$).  When you multiply an exponent by its base, the exponent increases (add the exponents: $x^2 \times x = x^3$ and $y^3 \times y^4 = y^7$), learn more in the Multiply Exponents lesson.
3. When numbers or variables are next to each other, they are related through multiplication, so $7x = 7 \times x$ and $ xy= x\times y$. When terms in parentheses are placed next to each other, they should be multipled, so $(7x)(6xy)=(7x)\times(6xy)=42x^2y$.
4. Whenever you multiply numbers, multiply them together, even if there are also variables involved. So, again $(7x)(6xy)=(7x)\times(6xy)=42x^2y$.
5. When you multiply two different variables (or a variable and a number) and you can't actually do the math (because you don't know what the variables are equal to!), you just set the variables next to each other (so, again $xy=x\times y$).

So, how do you multiply variable expressions?

Let's walk through the process when you multiply different combinations of numbers and variables.

When you multiply two numbers, you use multiplication to find a new number:

$3 \times 4= 24$

When you multiply a number times a variable, you can't actually do the math, so you just put the terms together:

$3 \times x= 3x$

When you multiply terms that contain numbers and variables, you do the math for the numbers and consolidate the like terms using exponents; unlike terms just get written in:

$$\eqalign{3x \times 4x &= 3 \times 4 \times x \times x\\&=12 \times x \times x\\&=12\times x^2\\&=12x^2}$$

or

$$\eqalign{5x \times 2y &= 5 \times 2 \times x \times y\\&=10 \times x \times y\\&=10\times xy\\&=10xy}$$

When you multiply terms that contain numbers and variables, simplify and consolidate terms first (numbers with numbers and like variables with like variables), then multiply:

$$\eqalign{(8abx^2)(5cdx)&= 8 \times 5 \times x^2 \times x \times a \times b \times c \times d\\&=40 \times x^3 \times a \times b \times c \times d\\&=40\times x^3 \times abcd\\&=40x^3abcd}$$

Note: in the final answer, variables should be written with the variable raised to the highest power first (if more than one variable is raise to the same power, they should be in alphabetical order).

You can multiply as many variable expressions as you need to this way (combine like terms, multiply numbers, raise variables to powers):

$$\eqalign{(3bx^2)(5cx)(2ab)(4xy)&= 3 \times 5 \times 2 \times 4\times x^2 \times x \times x \times b \times b \times a \times c\times y\\&=100 \times x^4 \times b \times b \times a\times c \times y\\&=120\times x^4 \times b^2 \times acy\\&=120x^4b^2acy}$$