Rules of Exponents

Learning Objectives

• Understand rules of exponents.
• Simplify and solve expressions in exponential notation.

We need a common language in order to communicate mathematical ideas clearly and efficiently. Exponential notationA condensed way of expressing repeated multiplication of a value by itself. Exponential notation consists of a base and an exponent. In the exponential term `5^3`, `5` is the base and `3` is the exponent. Exponential notation is a shorthand way of writing `5 * 5 * 5`. Also called exponential form. is one example. It was developed to express repeated multiplication and to make it easier to write very large and very small numbers. For example, growth models for populations often use exponents to manage and manipulate large numbers that change quickly over time.

In order to work with exponents, we need to “speak the language” and learn a few rules first.

Exponential notation has two parts. The baseThe value that is raised to a power when a number is written in exponential notation. In the term `5^3`, `5` is the base and `3` is the exponent. , as the name suggests, is the number on the bottom. The other part of the notation is a small number written in superscript to the right of the base, called the exponentThe value that indicates the number of times another value is multiplied by itself in exponential notation. The exponent, also called the power, is written in superscript. In the term `5^3`, `5` is the base and `3` is the exponent. . Below are some examples of exponential notation. We’ll use these examples to learn about the notation.

Let’s start with `10^3`. The base is `10`. This means that `10` is a factorFor any number `x`, the numbers that can be evenly divided into `x` are called factors of `x`. For example, the number `20` has the factors `1`, `2`, `4`, `5`, `10`, and `20`. , and it’s going to be multiplied by itself some number of times. The precise number of times is given by the exponent, the number in superscript. In this case, the exponent is `3`, which means the base of `10` will be used as a factor `3` times. So `10^3` means `10*10*10`. 

Now we know what `10^3` means, but how do we pronounce it? We have a lot of choices: this term could be said as “`10` raised to the third power” or “`10` to the third,” or “`10` cubed.” The words “raised to a powerA way of describing the exponent in exponential notation. We can say the base is “raised to the power” of the exponent. For example, we can read `x^5` as “`x` raised to the power of `5`” or as “`x` raised to the `5`th power.” ” are inserted between the base and the exponent to indicate exponential notation.

Okay then. Let’s consider `25^1`. What does an exponent of `1` mean? Any value raised to a power of `1` is just the value. This makes sense when we think about it, because the exponent of `1` means the base is used as a factor only once. So the base stands alone, and `25^1` is simply `25`.

That leaves us with the term `-3^4`. This example is a little trickier because there is a negative sign in there. One of the rules of exponential notation is that the exponent relates only to the value immediately to its left. So, `-3^4` does not mean `-3*-3*-3*-3`. It means “the opposite of `3^4`,” or  `-(3*3*3*3)`. If we wanted the base to be `-3`, we’d have to use parentheses in the notation: `(-3)^4`. Why so picky? Well, do the math:

`-3^4=-(3*3*3*3)=-81`

`(-3)^4 = -3*-3*-3*-3=81`

That’s an important difference.

We just learned the rule that the exponent only relates to the number directly to the left unless parentheses are used. Here’s another rule—when an exponent is present outside parentheses, everything inside is raised to that power. Consider the following example:

`(5 + 3)^2`

According to the order of operations, we must first simplify what is in the parentheses before we do any other operations. So we add `5` and `3` and then square the sum, `8`, to arrive at an answer of `64`. Another way to proceed is to rewrite `(5 + 3)^2` as `(5 + 3)(5 + 3)`, and then multiply it out to again get `64`.

`(5 + 3)^2 = (8)^2 = 8*8=64`

`(5 + 3)^2 = (5 + 3)(5 + 3) = 5(5 + 3) + 3(5+3) = 25 + 15 + 15 + 9 = 64`

Parentheses can be used in other ways with exponential notation. For example, we can use them to describe an exponential term to a power. For example, let’s take `5^2` and raise it to the `4`^th power. We’d write that as `(5^2)^4`. When a number written in exponential notation is raised to a power, it is called a “power of a power.”

In this expression, the base is `5^2` and the exponent is `4`, meaning `5^2` is to be used as a factor `4` times. We could rewrite this problem as `5^2*5^2*5^2*5^2` or `(5*5)*(5*5)*(5*5)*(5*5)`. Notice that works out to `5` multiplied `8` times. What’s another way to write that? `5^8`.

That leads us to another rule. Compare `5^8` to the original term of `(5^2)^4`. Notice that the new exponent is the same as the product of the original exponents: `2*4=8`. A shortcut for simplifying the power of a power is to multiply the exponents and keep the base the same.

There’s also a rule for combining two numbers in exponential form that have the same base. Consider the following expression:

`(2^3)(2^4)`

This can be rewritten as `(2*2*2)(2*2*2*2)` or `2*2*2*2*2*2*2`. In exponential form, you would write the product as `2^7`. Notice `7` is the sum of the original two exponents, `3` and `4`. To multiply exponential terms with the same base, add the exponents.

Exponents are not always positive. But what does it mean when an exponent is `0` or a negative integer? Let’s use what we know about powers of `10` to find out. 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. There’s a pattern there—see it?

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.

We can continue this pattern to add some more rows as shown below, each time dividing the number in one row by `10` to get the number in the next row:

Following the pattern, we see that `10^0` is equal to `1`. Then we get into negative exponents: `10^-1` is equal to `1/10`, and `10^-2` is the same as `1/100`. Now that’s interesting. Look back at the table, and see what `10` is in exponential form. It’s `10^1`. If we substitute that form of `10` into the fraction `1/10`, the fraction becomes `1/10^1`. So `10^-1=1/10^1`. Something very similar can be done with `10^(-2)`:

`10^-2=1/100` and `100 = 10^2`

`10^-2=1/10^2`

How about that? Numbers with negative exponents can be rewritten as a fraction, and not just any fraction. A number raised to a negative power is equivalent to the reciprocal of the number raised to the opposite of the power. That sounds complicated, but all it means is what we’ve just seen. A number to a negative power is the same as `1` over the number to the same but positive power. 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 a table with powers of `3`:

Yes, it all looks the same. The numbers are different but the patterns are the same. Now we know how numbers with zero and negative exponents behave.

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.

To multiply two terms with the same base, add their exponents. `(n^x)(n^y)=n^(x+y)`

To raise a power to a power, multiply the exponents. `(n^x)^y=n^(xy)`

For any nonzero number `n`, `n^0=1`.

For any nonzero number `n` and any integer `x`, `n^-x=1/n^x`.