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

In this module, you will learn the short-cut to writing 2 2 2 2 size 12{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 size 12{a rSup { size 8{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 a a a size 12{a rSup { size 8{n} } =1 cdot a cdot a cdot dotslow cdot a } {} ( n times)

with a multiplied by itself n times.

The resulting value is called the argument .

For example, instead of 5 5 5 5 5 5 size 12{5 cdot 5 cdot 5 cdot 5 cdot 5 cdot 5} {} , we write 5 6 size 12{5 rSup { size 8{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 size 12{5 rSup { size 8{2} } } {} is 5 5 size 12{5 cdot 5} {} and 3 5 size 12{3 rSup { size 8{5} } } {} is 3 3 3 3 3 size 12{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 ÷ ÷ a size 12{a rSup { size 8{ - n} } `=`1` div `a` div `a` div ` dotslow ` div `a } {}    ( n times)

If n is an even integer, then a n size 12{a rSup { size 8{n} } } {} will always be positive for any non-zero real number a . For example, although -2 is negative, ( 2 ) 2 = 1 2 2 = 4 size 12{ \( - 2 \) rSup { size 8{2} } =1 cdot - 2 cdot - 2=4} {}   is positive and so is ( 2 ) 2 = 1 ÷ 2 ÷ 2 = 1 4 size 12{ \( - 2 \) rSup { size 8{ - 2} } =1 div ` - 2 div ` - 2= { { size 6{ size 10{1}} } over { size 10{4}} } } {} .

Examples, exponential notation

Write the following multiplication using exponents:

3 · 3 

Since the factor 3 appears 2 times, we write this as

3 2

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62 · 62 · 62 · 62 · 62 · 62 · 62 · 62 · 62 

Since the factor 62 appears nine times, we write this as:

62 9

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Expand each number (write without exponents):

12 4 .        The exponent 4 indicates that the base (12) is repeated 4 times, thus:

12 4 = 12 · 12 · 12 · 12

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706 3 .   The exponent 3 indicates that the base (706) is repeated 3 times in a multiplication.

706 3 = 706 · 706 · 706

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Exercises, exponential notation

Write each of the following using exponents:

16 · 16 · 16 · 16 · 16

16 5

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9 · 9 · 9 · 9 · 9 · 9 · 9 · 9 · 9 · 9

9 10

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Write each of the following numbers without exponents:

4 7

4 · 4 · 4 · 4 · 4 · 4 · 4

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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.

a 0 = 1 size 12{a rSup { size 8{0} } =1} {}
a m × a n = a m + n size 12{a rSup { size 8{m} } times a rSup { size 8{n} } =a rSup { size 8{m+n} } } {}
a m ÷ a n = a m n size 12{a rSup { size 8{m} } ` div `a rSup { size 8{n} } `=`a rSup { size 8{m - n} } } {}
a n = 1 a n size 12{a rSup { size 8{ - n} } = { {1} over {a rSup { size 8{n} } } } } {}
ab n = a n b n size 12{ ital "ab" rSup { size 8{n} } =a rSup { size 8{n} } b rSup { size 8{n} } } {}
( a m ) n = a mn size 12{ \( a rSup { size 8{m} } \) rSup { size 8{n} } =a rSup { size 8{ ital "mn"} } } {}

We explain each law in detail in the following sections.

Exponential law 1

Our definition of exponential notation shows that:

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Source:  OpenStax, Basic math textbook for the community college. OpenStax CNX. Jul 04, 2009 Download for free at http://cnx.org/content/col10726/1.1
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