Exponential Notation

Learning Objectives

• Evaluate expressions containing exponents.
• Evaluate exponential notations with exponents of `0` and `1`.
• Write an exponential expression involving negative exponents with positive exponents.

A common language is needed in order to communicate mathematical ideas clearly and efficiently. Exponential notation A shorter way to write repeated multiplication. For example, `2^4` means `2 * 2 * 2 * 2`. Two is used as a factor `4` times. is one example. It was developed to write repeated multiplication more efficiently. For example, growth occurs in living organisms by the division of cells. One type of cell divides `2` times in an hour. So in `12` hours, the cell will divide `2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2` times. This can be written more efficiently as `2^12`.

We use exponential notation to write repeated multiplication, such as `10 * 10 * 10` as `10^3`. The `10` in `10^3` is called the base The expression that is being raised to a power when using exponential notation. In `5^3`, `5` is the base (which is the number that is repeatedly multiplied). `5^3 = 5 * 5 * 5`. And in `a^b`, the base is `a`. . The `3` in `10^3` is called the exponent When a number is expressed in the form `a^b`, `b` is the exponent. The exponent indicates how many times the base is used as a factor. Power and exponent mean the same thing. . The expression `10^3` is called the exponential expression.

`10^3` is read as “`10` to the third power” or “`10` raised to the power of `3`” or “`10` cubed.” It means `10 * 10 * 10`, or `1,000`.

`8^2` is read as “`8` to the second power” or “`8` raised to the power of `2`” or “`8` squared.” It means `8 * 8`, or `64`.

`5^4` is read as “`5` to the fourth power” or “`5` raised to the power of `4`.” It means `5 * 5 * 5 * 5`, or `625`.

`b^5` is read as “ `b` to the fifth power” or “`b` raised to the power of `5`.” It means `b * b * b * b * b`. Its value will depend on the value of `b`.

The exponent applies only to the number that it is next to. So in the expression `xy^4`, only the `y` is affected by the `4`. `xy^4` means `x * y * y * y * y`.

If the exponential expression is negative, such as `-3^4`, it means `-(3 * 3 * 3 * 3)` or `-81`.

If `-3` is to be the base of the exponential expression, it must be written as `(-3)^4`, which means `-3 * -3 * -3 * -3`, or `81`.

Likewise, `(-x)^4 = (-x) * (-x) * (-x) * (-x) = x^4`, while `-x^4 = -(x * x * x * x)`.

You can see that there is quite a difference, so you have to be very careful!

Evaluating expressions containing exponents is the same as evaluating any expression. You substitute the value of the variable into the expression and simplify.

You can use PEMDAS to remember the order in which you should evaluate the expression. First, evaluate anything in Parentheses or grouping symbols. Next, look for Exponents, followed by Multiplication and Division (reading from left to right), and lastly, Addition and Subtraction (again, reading from left to right).

So, when you evaluate the expression `5x^3` if `x = 4`, first substitute the value `4` for the variable `x`. Then evaluate, using order of operations.

Notice the difference between the example above and the one below.

The addition of parentheses made quite a difference!

What does it mean when an exponent is `0` or `1`? Let’s consider `25^1`. Any value raised to the power of `1` is just the value itself. This makes sense, because the exponent of `1` means the base is used as a factor A number or mathematical symbol that is multiplied by another number or mathematical symbol to form a product. For example, in the equation `4 * 5 = 20`, `4` and `5` are factors. only once. So the base stands alone, and `25^1` is simply `25`.

But what about a value raised to the power of `0`? Use what you know about powers of `10` to find out what the power of `0` means. Below is a list of powers of `10` and their equivalent values. Look at how the numbers change going down the left and right columns. Can you notice a pattern there?

Moving down the table, each row drops one factor of `10` from the one above it. From row `1` to row `2`, the exponential form goes from `10^5` to `10^4`. The value drops from `100,000` to `10,000`. Another way to put this is that each value is divided by `10` to produce the next value down the column.

Let’s use this pattern of division by `10` to predict the value of `10^0`.

Following the pattern, you see that `10^0` is equal to `1`. Would the pattern hold for a different base? Say a base of `3`?

Yes! And the same pattern would hold true for any non-zero number or variable raised to a power of `0`, `n^0 = 1`.

There is a conflict when the base is `0`. You know that `0^3 = 0`, `0^2 = 0`, and `0^1 = 0`, so you would expect `0^0` to also be equal to `0`. However, the above pattern says that any base raised to the power of `0` is `1`, so this leads you to believe that `0^0 = 1`. Notice the competing patterns: `0^0` cannot be both `0` and `1`! In this case, mathematicians say that the value of `0^0` is undefined. (And remember that undefined is not the same as `0`!)

As done previously, to evaluate expressions containing exponents of `0` or `1`, substitute the value of the variable into the expression and simplify.

What does it mean when an exponent is a negative integer? Let’s use the powers of `10` pattern from earlier to find out. If you continue this pattern to add some more rows, beyond `10^0`, you find the following:

Following the pattern, you see that `10^0` is equal to `1`. Then you get into negative exponents: `10^-1` is equal to `1/10^1`, and `10^-2` is the same as `1/10^2`.

Following this pattern, a number with a negative exponent can be rewritten as the reciprocal of the original number, with a positive exponent.

For example, `10^-3=1/10^3` and `10^-7=1/10^7`.

To see if these patterns hold true for numbers other than `10`, check out this table with powers of `3`.

The numbers are different but the patterns are the same. We are now ready to state the definition of a negative exponent.

Note that the definition above states that the base, `n` must be a “non-zero number.”

Exponential notation is composed of a base and an exponent. It is a “shorthand” way of writing repeated multiplication, and indicates that the base is a factor and the exponent is the number of times the factor is used in the multiplication. The basic rules of exponents are as follows:

• An exponent applies only to the value to its immediate left.
• When a quantity in parentheses is raised to a power, the exponent applies to everything inside the parentheses.
• For any non-zero number `n`, `n^0=1`.
• For any non-zero number `n` and any integer `x`, `n^-x=1/n^x`.