3.1: Exponents and Roots
- 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
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
\(8 \cdot 8 \cdot 8 \cdot 8 \cdot 8\)
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 .
An exponent records the number of identical factors that are repeated in a multiplication.
Write the following multiplication using exponents.
\(3 \cdot 3\). Since the factor 3 appears 2 times, we record this as
Solution
\(3^2\)
\(62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62 \cdot 62\). Since the factor 62 appears 9 times, we record this as
Solution
\(62^9\)
Expand (write without exponents) each number.
\(12^4\). The exponent 4 is recording 4 factors of 12 in a multiplication. Thus,
Solution
\(12^4 = 12 \cdot 12 \cdot 12 \cdot 12\)
\(706^3\). 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).
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: \(5 \cdot 5 = 5^2 = 25\)
\(\sqrt[5]{32}\) 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.
Use the calculator to find \(\sqrt{121}\)
Solution
| Display Reads | ||
| Type | 121 | 121 |
| Press | \(\sqrt{x}\) | 11 |
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.