We need a common language in order to communicate mathematical ideas clearly and efficiently.

Exponential notation is one example.

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It was developed to express repeated multiplication and to make it easier to write very long 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.

**What is Exponential Notation?**

Exponential notation has two parts. The **base**, 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 **exponent**. Below are some examples of exponential notation. We’ll use these examples to learn about the notation.

10^{3} | 25^{1} | -3^{4} |

Let’s start with 10^{3}.^{ }The base is 10. This means that 10 is a **factor**, 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 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 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.

**Rules for Computing with Exponents**

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

Parenthesis can be used in others 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 (52)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: 5^{2 }is to be used as a factor 4 times. We could rewrite this problem 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.

**Rules of Exponents**

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*^{x}^{y}