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8.3: Exponents and Roots

  • Page ID
    116801
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    Learning Objectives
    • understand and be able to read exponential notation
    • understand the concept of root and be able to read root notation
    • be able to use a calculator having the \(y^x\) key to determine a root

    Exponential Notation

    Definition: 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

    \(8 \cdot 8 \cdot 8 \cdot 8 \cdot 8\)

    Definition: 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.

    Definition: 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.

    \(3 \cdot 3\). Since the factor 3 appears 2 times, we record this as

    Solution

    \(3^2\)

    Sample Set A

    \(62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62\). Since the factor 62 appears 9 times, we record this as

    Solution

    \(62^9\)

    Expand (write without exponents) each number.

    Sample Set A

    \(12^4\). The exponent 4 is recording 4 factors of 12 in a multiplication. Thus,

    Solution

    \(12^4 = 12 \cdot 12 \cdot 12 \cdot 12\)

    Sample Set A

    \(706^3\). The exponent 3 is recording 3 factors of 706 in a multiplication. Thus,

    Solution

    \(706^3 = 706 \cdot 706 \cdot 706\)

    Practice Set A

    Write the following using exponents.

    \(37 \cdot 37\)

    Answer

    \(37^2\)

    Practice Set A

    \(16 \cdot 16 \cdot 16 \cdot 16 \cdot 16\)

    Answer

    \(16^5\)

    Practice Set A

    \(9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9 \cdot 9\)

    Answer

    \(9^{10}\)

    Write each number without exponents.

    Practice Set A

    \(85^3\)

    Answer

    \(85 \cdot 85 \cdot 85\)

    Practice Set A

    \(4^7\)

    Answer

    \(4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4 \cdot 4\)

    Practice Set A

    \(1,739^2\)

    Answer

    \(1,739 \cdot 1,739\)

    Reading Exponential Notation

    In a number such as \(8^5\).

    Base
    8 is called the base.

    Exponent, Power
    5 is called the exponent, or power. \(8^5\) 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 \(5^2\) 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 \(5^3\) 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 \(5^8\) 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 \(49 = 7^2\), that is, \(49 = 7 \cdot 7\). 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 \(8 = 2^3\), that is, \(8 = 2 \cdot 2 \cdot 2\). 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.

    We can continue this way to see such roots as fourth roots, fifth roots, sixth roots, and so on.

    Reading Root Notation

    There is a symbol used to indicate roots of a number. It is called the radical sign \(\sqrt[n]{\ \ \ \ }\)

    The Radical Sign \(\sqrt[n]{\ \ \ \ }\)

    The symbol \(\sqrt[n]{\ \ \ \ }\) is called a radical sign and indicates the nth root of a number.

    We discuss particular roots using the radical sign as follows:

    Square Root
    \(\sqrt[2]{\text{number}}\) indicates the square root of the number under the radical sign. It is customary to drop the 2 in the radical sign when discussing square roots. The symbol \(\sqrt{\ \ \ \ }\) is understood to be the square root radical sign.
    \(\sqrt{49} = 7\) since \(7 \cdot 7 = 7^2 = 49\)

    Cube Root
    \(\sqrt[3]{\text{number}}\) indicates the cube root of the number under the radical sign.
    \(\sqrt[3]{8} = 2\) since \(2 \cdot 2 \cdot 2 = 2^3 = 8\)

    Fourth Root
    \(\sqrt[4]{\text{number}}\) indicates the fourth root of the number under the radical sign.
    \(\sqrt[4]{81} = 3\) since \(3 \cdot 3 \cdot 3 \cdot 3 = 3^4 = 81\)
    In an expression such as \(\sqrt[5]{32}\)

    Radical Sign
    \(\sqrt{\ \ \ \ }\) is called the radical sign.

    Index
    5 is called the index. (The index describes the indicated root.)

    Radicand
    32 is called the radicand.

    Radical
    \(\sqrt[5]{32}\) is called a radical (or radical expression).

    Sample Set B

    Find each root.

    \(\sqrt{25}\) To determine the square root of 25, we ask, "What whole number squared equals 25?" From our experience with multiplication, we know this number to be 5. Thus,

    Solution

    \(\sqrt{25} = 5\)

    Check: \(5 \cdot 5 = 5^2 = 25\)

    Sample Set B

    \(\sqrt[5]{32}\) To determine the fifth root of 32, we ask, "What whole number raised to the fifth power equals 32?" This number is 2.

    Solution

    \(\sqrt[5]{32} = 2\)

    Check: \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^5 = 32\)

    Practice Set B

    Find the following roots using only a knowledge of multiplication.

    \(\sqrt{64}\)

    Answer

    8

    Practice Set B

    \(\sqrt{100}\)

    Answer

    10

    Practice Set B

    \(\sqrt[3]{64}\)

    Answer

    4

    Practice Set B

    \(\sqrt[6]{64}\)

    Answer

    2

    Calculators

    Calculators with the \(\sqrt{x}\), \(y^x\), and \(1/x\) keys can be used to find or approximate roots.

    Sample Set C

    Use the calculator to find \(\sqrt{121}\)

    Solution

        Display Reads
    Type 121 121
    Press \(\sqrt{x}\) 11
    Sample Set C

    Find \(\sqrt[7]{2187}\).

    Solution

        Display Reads
    Type 2187 2187
    Press \(y^x\) 2187
    Type 7 7
    Press \(1/x\) .14285714
    Press = 3

    \(\sqrt[3]{2187} = 3\) (which means that \(3^7 = 2187\))

    Practice Set C

    Use a calculator to find the following roots.

    \(\sqrt[3]{729}\)

    Answer

    9

    Practice Set C

    \(\sqrt[4]{8503056}\)

    Answer

    54

    Practice Set C

    \(\sqrt{53361}\)

    Answer

    231

    Practice Set C

    \(\sqrt[12]{16777216}\)

    Answer

    4

    Exercises

    For the following problems, write the expressions using expo­nential notation.

    Exercise \(\PageIndex{1}\)

    \(4 \cdot 4\)

    Answer

    \(4^2\)

    Exercise \(\PageIndex{2}\)

    \(12 \cdot 12\)

    Exercise \(\PageIndex{3}\)

    \(9 \cdot 9 \cdot 9 \cdot 9\)

    Answer

    \(9^4\)

    Exercise \(\PageIndex{4}\)

    \(10 \cdot 10 \cdot 10 \cdot 10 \cdot 10 \cdot 10\)

    Exercise \(\PageIndex{5}\)

    \(826 \cdot 826 \cdot 826\)

    Answer

    \(826^3\)

    Exercise \(\PageIndex{6}\)

    \(3,021 \cdot 3,021 \cdot 3,021 \cdot 3,021 \cdot 3,021\)

    Exercise \(\PageIndex{7}\)

    \(\begin{matrix} \underbrace{6 \cdot 6 \cdots\cdots 6} \\ {\text{85 factors of 6}} \end{matrix}\)

    Answer

    \(6^{85}\)

    Exercise \(\PageIndex{8}\)

    \(\begin{matrix} \underbrace{2 \cdot 2 \cdots\cdots 2} \\ {\text{112 factors of 2}} \end{matrix}\)

    Exercise \(\PageIndex{9}\)

    \(\begin{matrix} \underbrace{1 \cdot 1 \cdots\cdots 1} \\ {\text{3,008 factors of 1}} \end{matrix}\)

    Answer

    \(1^{3008}\)

    For the following problems, expand the terms. (Do not find the actual value.)

    Exercise \(\PageIndex{10}\)

    \(5^3\)

    Exercise \(\PageIndex{11}\)

    \(7^4\)

    Answer

    \(7 \cdot 7 \cdot 7 \cdot 7\)

    Exercise \(\PageIndex{12}\)

    \(15^2\)

    Exercise \(\PageIndex{13}\)

    \(117^5\)

    Answer

    \(117 \cdot 117 \cdot 117 \cdot 117 \cdot 117\)

    Exercise \(\PageIndex{14}\)

    \(61^6\)

    Exercise \(\PageIndex{15}\)

    \(30^2\)

    Answer

    \(30 \cdot 30\)

    For the following problems, determine the value of each of the powers. Use a calculator to check each result.

    Exercise \(\PageIndex{16}\)

    \(3^2\)

    Exercise \(\PageIndex{17}\)

    \(4^2\)

    Answer

    \(4 \cdot 4 = 16\)

    Exercise \(\PageIndex{18}\)

    \(1^2\)

    Exercise \(\PageIndex{19}\)

    \(10^2\)

    Answer

    \(10 \cdot 10 = 100\)

    Exercise \(\PageIndex{20}\)

    \(11^2\)

    Exercise \(\PageIndex{21}\)

    \(12^2\)

    Answer

    \(12 \cdot 12 = 144\)

    Exercise \(\PageIndex{22}\)

    \(13^2\)

    Exercise \(\PageIndex{23}\)

    \(15^2\)

    Answer

    \(15 \cdot 15 = 225\)

    Exercise \(\PageIndex{24}\)

    \(1^4\)

    Exercise \(\PageIndex{25}\)

    \(3^4\)

    Answer

    \(3 \cdot 3 \cdot 3 \cdot 3 = 81\)

    Exercise \(\PageIndex{26}\)

    \(7^3\)

    Exercise \(\PageIndex{27}\)

    \(10^3\)

    Answer

    \(10 \cdot 10 \cdot 10 = 1000\)

    Exercise \(\PageIndex{28}\)

    \(100^2\)

    Exercise \(\PageIndex{29}\)

    \(8^3\)

    Answer

    \(8 \cdot 8 \cdot 8 = 512\)

    Exercise \(\PageIndex{30}\)

    \(5^5\)

    Exercise \(\PageIndex{31}\)

    \(9^3\)

    Answer

    \(9 \cdot 9 \cdot 9 = 729\)

    Exercise \(\PageIndex{32}\)

    \(6^2\)

    Exercise \(\PageIndex{33}\)

    \(7^1\)

    Answer

    \(7^1 = 7\)

    Exercise \(\PageIndex{34}\)

    \(1^{28}\)

    Exercise \(\PageIndex{35}\)

    \(2^7\)

    Answer

    \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 128\)

    Exercise \(\PageIndex{36}\)

    \(0^5\)

    Exercise \(\PageIndex{37}\)

    \(8^4\)

    Answer

    \(8 \cdot 8 \cdot 8 \cdot 8 = 4,096\)

    Exercise \(\PageIndex{38}\)

    \(5^8\)

    Exercise \(\PageIndex{39}\)

    \(6^9\)

    Answer

    \(6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6 = 10,077,696\)

    Exercise \(\PageIndex{40}\)

    \(25^3\)

    Exercise \(\PageIndex{41}\)

    \(42^2\)

    Answer

    \(42 \cdot 42 = 1,764\)

    Exercise \(\PageIndex{42}\)

    \(31^3\)

    Exercise \(\PageIndex{43}\)

    \(15^5\)

    Answer

    \(15 \cdot 15 \cdot 15 \cdot 15 \cdot 15 = 759,375\)

    Exercise \(\PageIndex{44}\)

    \(2^{20}\)

    Exercise \(\PageIndex{45}\)

    \(816^2\)

    Answer

    \(816 \cdot 816 = 665,856\)

    For the following problems, find the roots (using your knowledge of multiplication). Use a calculator to check each result.

    Exercise \(\PageIndex{46}\)

    \(\sqrt{9}\)

    Exercise \(\PageIndex{47}\)

    \(\sqrt{16}\)

    Answer

    4

    Exercise \(\PageIndex{48}\)

    \(\sqrt{36}\)

    Exercise \(\PageIndex{49}\)

    \(\sqrt{64}\)

    Answer

    8

    Exercise \(\PageIndex{50}\)

    \(\sqrt{121}\)

    Exercise \(\PageIndex{51}\)

    \(\sqrt{144}\)

    Answer

    12

    Exercise \(\PageIndex{52}\)

    \(\sqrt{169}\)

    Exercise \(\PageIndex{53}\)

    \(\sqrt{225}\)

    Answer

    15

    Exercise \(\PageIndex{54}\)

    \(\sqrt[3]{27}\)

    Exercise \(\PageIndex{55}\)

    \(\sqrt[5]{32}\)

    Answer

    2

    Exercise \(\PageIndex{56}\)

    \(\sqrt[7]{1}\)

    Exercise \(\PageIndex{57}\)

    \(\sqrt{400}\)

    Answer

    20

    Exercise \(\PageIndex{58}\)

    \(\sqrt{900}\)

    Exercise \(\PageIndex{59}\)

    \(\sqrt{10,000}\)

    Answer

    100

    Exercise \(\PageIndex{60}\)

    \(\sqrt{324}\)

    Exercise \(\PageIndex{61}\)

    \(\sqrt{3,600}\)

    Answer

    60

    For the following problems, use a calculator with the keys \(\sqrt{x}\), \(y^x\), and \(1/x\) to find each of the values.

    Exercise \(\PageIndex{62}\)

    \(\sqrt{676}\)

    Exercise \(\PageIndex{63}\)

    \(\sqrt{1,156}\)

    Answer

    34

    Exercise \(\PageIndex{64}\)

    \(\sqrt{46,225}\)

    Exercise \(\PageIndex{65}\)

    \(\sqrt{17,288,964}\)

    Answer

    4,158

    Exercise \(\PageIndex{66}\)

    \(\sqrt[3]{3,375}\)

    Exercise \(\PageIndex{67}\)

    \(\sqrt[4]{331,776}\)

    Answer

    24

    Exercise \(\PageIndex{68}\)

    \(\sqrt[8]{5,764,801}\)

    Exercise \(\PageIndex{69}\)

    \(\sqrt[12]{16,777,216}\)

    Answer

    4

    Exercise \(\PageIndex{70}\)

    \(\sqrt[8]{16,777,216}\)

    Exercise \(\PageIndex{71}\)

    \(\sqrt[10]{9,765,625}\)

    Answer

    5

    Exercise \(\PageIndex{72}\)

    \(\sqrt[4]{160,000}\)

    Exercise \(\PageIndex{73}\)

    \(\sqrt[3]{531,441}\)

    Answer

    81

    Exercises for Review

    Exercise \(\PageIndex{74}\)

    Use the numbers 3, 8, and 9 to illustrate the associative property of addition.

    Exercise \(\PageIndex{75}\)

    In the multiplication \(8 \cdot 4 = 32\), specify the name given to the num­bers 8 and 4.

    Answer

    81

    Exercise \(\PageIndex{76}\)

    Does the quotient \(15 \div 0\) exist? If so, what is it?

    Exercise \(\PageIndex{77}\)

    Does the quotient \(0 \div 15\) exist? If so, what is it?

    Answer

    Yes; 0

    Exercise \(\PageIndex{78}\)

    Use the numbers 4 and 7 to illustrate the commutative property of multiplication.


    This page titled 8.3: Exponents and Roots is shared under a CC BY license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .