Basic Math and Pre-Algebra For Dummies (98 page)

As algebraic expressions grow more complex, simplifying them can make them easier to work with. Simplifying an expression means (quite simply!) making it smaller and easier to manage. You see how important simplifying expressions becomes when you begin solving algebraic equations.

For now, think of this section as a kind of algebra toolkit. Here I show you
how
to use these tools. In Chapter
22
, I show you
when
to use them.

When two algebraic terms contain like terms (when their variables match), you can add or subtract them (see the earlier section “Considering algebraic terms and the Big Four”). This feature comes in handy when you're trying to simplify an expression. For example, suppose you're working with the following expression:

• 4
  x
  – 3
  y
  + 2
  x
  +
  y
  –
  x
  + 2
  y
  

As it stands, this expression has six terms. But three terms have the variable
x
and the other three have the variable
y.
Begin by rearranging the expression so that all like terms are grouped together:

• = 4
  x
  + 2
  x
  –
  x
  – 3
  y
  +
  y
  + 2
  y
  

Now you can add and subtract like terms. I do this in two steps, first for the
x
terms and then for the
y
terms:

Notice that the
x
terms simplify to 5
x,
and the
y
terms simplify to 0
y,
which is 0, so the
y
terms drop out of the expression altogether.

Here's a somewhat more complicated example that has variables with exponents:

This time, you have four different types of terms. As a first step, you can rearrange these terms so that groups of like terms are all together (I underline these four groups so you can see them clearly):

Now combine each set of like terms:

This time, the
x
²
terms add up to 0, so they drop out of the expression altogether:

• = 5
  x
  + 9
  xy
  + 8
  y
  

Parentheses keep parts of an expression together as a single unit. In Chapter
5
, I show you how to handle parentheses in an arithmetic expression. This skill is also useful with algebraic expressions. As you find when you begin solving algebraic equations in Chapter
22
, getting rid of parentheses is often the first step toward solving a problem. In this section, I show how to handle the Big Four operations with ease.

When an expression contains parentheses that come right after a plus sign (+), you can just remove the parentheses. Here's an example:

Now you can simplify the expression by combining like terms:

• = 5
  x
  + 4
  y
  

When the first term inside the parentheses is negative, when you drop the parentheses, the minus sign replaces the plus sign. For example:

 Sometimes an expression contains parentheses that come right after a minus sign (–). In this case, change the sign of every term inside the parentheses to the opposite sign; then remove the parentheses.

Consider this example:

• 6
  x
  – (2
  xy
  – 3
  y
  ) + 5
  xy
  

A minus sign is in front of the parentheses, so you need to change the signs of both terms in the parentheses and remove the parentheses. Notice that the term 2
xy
appears to have no sign because it's the first term inside the parentheses. This expression really means the following:

• = 6
  x
  – (+2
  xy
  – 3
  y
  )
  + 5
  xy
  

You can see how to change the signs:

• = 6
  x
  – 2
  xy
  + 3
  y
  + 5
  xy
  

At this point, you can combine the two
xy
terms:

• = 6
  x
  + 3
  xy
  + 3
  y
  

When you see nothing between a number and a set of parentheses, it means multiplication. For example,

• 2(3) = 6

  4(4) = 16

  10(15) = 150

This notation becomes much more common with algebraic expressions, replacing the multiplication sign (×) to avoid confusion with the variable
x:

• 3(4 x ) = 12 x

  4 x (2 x ) = 8 x 2

  3 x (7 y ) = 21 xy

 To remove parentheses without a sign, multiply the term outside the parentheses by every term inside the parentheses; then remove the parentheses. When you follow those steps, you're using the
distributive property.

Here's an example:

• 2(3
  x
  – 5
  y
  + 4)
  

In this case, multiply 2 by each of the three terms inside the parentheses:

• = 2(3
  x
  ) + 2(–5
  y
  ) + 2(4)
  

For the moment, this expression looks more complex than the original one, but now you can get rid of all three sets of parentheses by multiplying:

• = 6
  x
  – 10
  y
  + 8
  

Multiplying by every term inside the parentheses is simply distribution of multiplication over addition — also called the
distributive property
— which I discuss in Chapter
4
.

As another example, suppose you have the following expression: