Basic Math and Pre-Algebra For Dummies (24 page)

So
.

Sometimes the entire contents of a set of parentheses are raised to an exponent. In this case, evaluate the contents of the parentheses
before
evaluating the exponent, as usual. Here's an example:

• (7 − 5)
  ³
  

First, evaluate 7 − 5:

• = 2
  ³
  

With the parentheses removed, you're ready to evaluate the exponent:

• = 8
  

Once in a rare while, the exponent itself contains parentheses. As always, evaluate what's in the parentheses first. For example,

This time, the smaller expression inside the parentheses is a mixed-operator expression. I've underlined the part that you need to evaluate first:

Now you can finish off what's inside the parentheses:

• = 21
  ¹
  

At this point, all that's left is a very simple exponent:

• = 21
  

So
.

Note:
Technically, you don't need to put parentheses around the exponent. If you see an expression in the exponent, treat it as though it has parentheses around it. In other words,
means the same as
.

Occasionally, an expression has
nested parentheses,
or one or more sets of parentheses inside another set. Here I give you the rule for handling nested parentheses.

 When evaluating an expression with nested parentheses, evaluate what's inside the
innermost
set of parentheses first and work your way toward the
outermost
parentheses.

For example, suppose you want to evaluate the following expression:

• 2 + (9 − (
  7
  −
  3
  ))
  

I underlined the contents of the innermost set of parentheses, so evaluate these contents first:

• = 2 + (9 − 4)
  

Next, evaluate what's inside the remaining set of parentheses:

• = 2 + 5
  

Now you can finish things off easily:

• = 7
  

So 2 + (9 − (7 − 3)) = 7.

As a final example, here's an expression that requires everything from this chapter:

This expression is about as complicated as you're ever likely to see in pre-algebra: one set of parentheses containing another set, which contains a third set. To start you off, I underlined what's deep inside this third set of parentheses. This is where you begin evaluating:

What's left is one set of parentheses inside another set. Again, work from the inside out. The smaller expression here is
, so evaluate the exponent first, then the multiplication, and finally the subtraction:

Only one more set of parentheses to go:

• = 4 + 56
  

At this point, finishing up is easy:

• = 60
  

Therefore,
.