1.1 Exponents

In this module, you will learn the short-cut to writing $2\cdot 2\cdot 2\cdot 2$ . This is known as writing a number in exponential notation .

Definition of exponential notation

Exponential notation is a short way of writing the same number multiplied by itself many times.

Exponential notation uses a superscript for the number of times the number is repeated. The superscript is placed on the number to be multiplied (the factor), and is written like $a^n$ where n is an integer and a can be any real number. a is called the base and n is called the exponent or power .

The n th power of a is defined as:

$a^n=1\cdot a\cdot a\cdot \dots \cdot a$ ( n times)

with a multiplied by itself n times.

The resulting value is called the argument .

For example, instead of $5\cdot 5\cdot 5\cdot 5\cdot 5\cdot 5$ , we write $5^6$ to show that the number 5 is multiplied by itself 6 times.

5 is the base, and 6 is the exponent or power.

The result, 15625, is the argument.

5 ⁶ is read as “five to the sixth power,” or more simply as “five to the sixth,” or “the sixth power of five.”

Likewise $5^2$ is $5\cdot 5$ and $3^5$ is $3\cdot 3\cdot 3\cdot 3\cdot 3$ . We will now have a closer look at writing numbers using exponential notation.

When a whole number is raised to the second power, it is said to be squared . The number 5 ² can be read as

• 5 to the second power, or
• 5 to the second, or
• 5 squared.

When a whole number is raised to the third power, it is said to be cubed . The number 5 ³ can be read as

• 5 to the third power, or
• 5 to the third, or
• 5 cubed.

When a whole number is raised to the power of 4 or higher, we simply say that the number is raised to that particular power. The number 5 ⁸ can be read as

• 5 to the eighth power, or just
• 5 to the eighth.

We can also define what it means if we have a negative index, - n . Then,

$a^{-n}=1÷a÷a÷\dots ÷a$    ( n times)

If n is an even integer, then $a^n$ will always be positive for any non-zero real number a . For example, although -2 is negative, $(-2{)}^2=1\cdot -2\cdot -2=4$   is positive and so is $(-2{)}^{-2}=1÷-2÷-2=\frac{1}{4}$ .

Examples, exponential notation

Write the following multiplication using exponents:

Expand each number (write without exponents):

Exercises, exponential notation

Write each of the following using exponents:

Write each of the following numbers without exponents:

Laws of exponents

There are several laws we can use to make working with exponential numbers easier. We list all the laws here for easy reference.

We explain each law in detail in the following sections.

Exponential law 1

Our definition of exponential notation shows that: