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3.1 Exponents and roots

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses exponents and roots. By the end of the module students should be able to understand and be able to read exponential notation, understand the concept of root and be able to read root notation, and use a calculator having the y x key to determine a root.

Section overview

  • Exponential Notation
  • Reading Exponential Notation
  • Roots
  • Reading Root Notation
  • Calculators

Exponential notation

Exponential notation

We have noted that multiplication is a description of repeated addition. Exponen­tial notation is a description of repeated multiplication.

Suppose we have the repeated multiplication

8 ⋅ 8 ⋅ 8 ⋅ 8 ⋅ 8 size 12{8 cdot 8 cdot 8 cdot 8 cdot 8} {}

Exponent

The factor 8 is repeated 5 times. Exponential notation uses a superscript for the number of times the factor is repeated. The superscript is placed on the repeated factor, 8 5 , in this case. The superscript is called an exponent .

The function of an exponent

An exponent records the number of identical factors that are repeated in a multiplication.

Sample set a

Write the following multiplication using exponents.

3 ⋅ 3 size 12{3 cdot 3} {} . Since the factor 3 appears 2 times, we record this as

3 2 size 12{3 rSup { size 8{2} } } {}

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62 ⋅ 62 ⋅ 62 ⋅ 62 ⋅ 62 ⋅ 62 ⋅ 62 ⋅ 62 ⋅ 62 size 12{"62" cdot "62" cdot "62" cdot "62" cdot "62" cdot "62" cdot "62" cdot "62" cdot "62"} {} . Since the factor 62 appears 9 times, we record this as

62 9 size 12{"62" rSup { size 8{9} } } {}

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Expand (write without exponents) each number.

12 4 size 12{"12" rSup { size 8{4} } } {} . The exponent 4 is recording 4 factors of 12 in a multiplication. Thus,

12 4 = 12 ⋅ 12 ⋅ 12 ⋅ 12 size 12{"12" rSup { size 8{4} } ="12" cdot "12" cdot "12" cdot "12"} {}

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706 3 size 12{"706" rSup { size 8{3} } } {} . The exponent 3 is recording 3 factors of 706 in a multiplication. Thus,

706 3 = 706 ⋅ 706 ⋅ 706 size 12{"706" rSup { size 8{3} } ="706" cdot "706" cdot "706"} {}

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Practice set a

Write the following using exponents.

37 ⋅ 37 size 12{"37" cdot "37"} {}

37 2 size 12{"37" rSup { size 8{2} } } {}

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16 ⋅ 16 ⋅ 16 ⋅ 16 ⋅ 16 size 12{"16" cdot "16" cdot "16" cdot "16" cdot "16"} {}

16 5 size 12{"16" rSup { size 8{5} } } {}

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9 ⋅ 9 ⋅ 9 ⋅ 9 ⋅ 9 ⋅ 9 ⋅ 9 ⋅ 9 ⋅ 9 ⋅ 9 size 12{9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9 cdot 9} {}

9 10 size 12{9 rSup { size 8{"10"} } } {}

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Write each number without exponents.

85 3 size 12{"85" rSup { size 8{3} } } {}

85 ⋅ 85 ⋅ 85 size 12{"85" cdot "85" cdot "85"} {}

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4 7 size 12{4 rSup { size 8{7} } } {}

4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 size 12{4 cdot 4 cdot 4 cdot 4 cdot 4 cdot 4 cdot 4} {}

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1, 739 2 size 12{1,"739" rSup { size 8{2} } } {}

1, 739 ⋅ 1, 739 size 12{1,"739" cdot 1,"739"} {}

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Reading exponential notation

In a number such as 8 5 size 12{8 rSup { size 8{5} } } {} ,

Base

8 is called the base .

Exponent, power

5 is called the exponent , or power . 8 5 size 12{8 rSup { size 8{5} } } {} is read as "eight to the fifth power," or more simply as "eight to the fifth," or "the fifth power of eight."

Squared

When a whole number is raised to the second power, it is said to be squared . The number 5 2 size 12{5 rSup { size 8{2} } } {} can be read as

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

Cubed

When a whole number is raised to the third power, it is said to be cubed . The number 5 3 size 12{5 rSup { size 8{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 that number is raised to that particular power. The number 5 8 size 12{5 rSup { size 8{8} } } {} can be read as

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

Roots

In the English language, the word "root" can mean a source of something. In mathematical terms, the word "root" is used to indicate that one number is the source of another number through repeated multiplication.

Square root

We know that 49 = 7 2 size 12{"49"=7 rSup { size 8{2} } } {} , that is, 49 = 7 ⋅ 7 size 12{"49"=7 cdot 7} {} . Through repeated multiplication, 7 is the source of 49. Thus, 7 is a root of 49. Since two 7's must be multiplied together to produce 49, the 7 is called the second or square root of 49.

Cube root

We know that 8 = 2 3 size 12{8=2 rSup { size 8{3} } } {} , that is, 8 = 2 ⋅ 2 ⋅ 2 size 12{8=2 cdot 2 cdot 2} {} . Through repeated multiplication, 2 is the source of 8. Thus, 2 is a root of 8. Since three 2's must be multiplied together to produce 8, 2 is called the third or cube root of 8.

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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