1.2 Use the language of algebra  (Page 3/18)

Suppose we need to multiply 2 nine times. We could write this as $2·2·2·2·2·2·2·2·2.$ This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write $2·2·2$ as $2^3$ and $2·2·2·2·2·2·2·2·2$ as $2^9.$ In expressions such as $2^3,$ the 2 is called the base and the 3 is called the exponent . The exponent tells us how many times we need to multiply the base.

We read $2^3$ as “two to the third power” or “two cubed.”

We say $2^3$ is in exponential notation and $2·2·2$ is in expanded notation .

Exponential notation

$a^n$ means multiply a by itself, n times.

The expression $a^n$ is read a to the $n^{th}$ power.

While we read $a^n$ as “ a to the $n^{th}$ power,” we usually read:

• $a^2$ “ a squared”
• $a^3$ “ a cubed”

We’ll see later why $a^2$ and $a^3$ have special names.

[link] shows how we read some expressions with exponents.

Simplify expressions using the order of operations

To simplify an expression    means to do all the math possible. For example, to simplify $4·2+1$ we’d first multiply $4·2$ to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:

By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.

We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values. For example, consider the expression:

If you simplify this expression, what do you get?

Some students say 49,

Others say 25,

Imagine the confusion in our banking system if every problem had several different correct answers!

The same expression should give the same result. So mathematicians early on established some guidelines that are called the Order of Operations .