3.4 Powers

Simplify Expressions with Exponents

To simplify a numerical 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:

$$4·2+1$$

$$8+1$$

$$9$$

Suppose we have the expression $2·2·2·2·2·2·2·2·2.$ We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. 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 factors of the base we have to multiply.

$$\text{means multiply three factors of 2}$$

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

For powers of $n=2$ and $n=3,$ we have special names.

Table 3.1 lists some examples of expressions written in exponential notation.

To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.

Practice Makes Perfect

Simplify Expressions with Exponents

In the following exercises, write in exponential form.

In the following exercises, write in expanded form.