In Definition 1.31, we introduced the concept of multiplication as repeated addition, and we build upon that idea here. We define exponentiation as repeated multiplication.
SubsectionDefinition of exponentiation
Definition1.54.
Let \(b\) be an integer and \(n\) be a positive integer. We define the \(n\)-th power of \(b\) to be the product of \(n\) copies of \(b\text{.}\)
We denote the \(n\)-th power of \(b\) by \(b^n\text{.}\) In short we also read \(b^n\) as “\(b\) to the \(n\)-th”.
We call \(b\) the base of \(b^n\) and \(n\) the exponent of \(b^n\text{.}\)
By Definition 1.54 an integer to the first power is the integer itself. That is for any integer \(b\) we have \(b^1=b\text{.}\)
Powers with the common exponents two and three are also read differently. Let \(b\) be an integer. We often read \(b^2\) as \(b\) squared (also see Definition 1.29) and \(b^3\) as “\(b\) cubed”.
In the video in Figure 1.55 the definition of exponentiation is followed by an overview of the topics in the remainder of this section.
For examples of powers we show how they are read.
Example1.56.Reading powers.
\(3^2=9\) is read “\(3\) squared is equal to \(9\)” or “3 to the 2-nd is equal to 9”
\(2^3=8\) is read “2 to the third is equal to 8”
\(2^4=16\) is read “2 to the 4th is equal to 16”
For examples of powers we identify the base and exponent.
Because \(b^m\) is the product of \(m\) copies of \(b\) and \(b^n\) is the product of \(n\) copies of \(b\) we have that \((b^m)\cdot (b^n)\) is the product of \(m+n\) copies of \(b\text{.}\) Writing the latter as a power we get
Because \({(b^m)}^n\) is the product of \(n\) copies of \(b^m\) and \(b^m\) is the product of \(m\) copies of \(b\) we have that \({(b^m)}^n\) is the product of \(m\cdot n\) copies of \(b\text{.}\) Writing the latter as a power we get
Use the properties of exponentiation to simplify \(1256^3\cdot 1256^{11}\text{.}\)
Solution.
We apply Theorem 1.59 which states that for all integers \(b\) and for all non-negative integers \(m\) and \(n\) we have \(b^m\cdot b^n=b^{m+n}\text{.}\) With \(b=1256\) and \(m=3\) and \(n=11\) we get
Let \(d\) be an integer. Use the properties of exponentiation to simplify \(d^9\cdot d^7\cdot d^3\text{.}\)
Solution.
We apply Theorem 1.59 which states that for non-negative integers \(m\) and \(n\) we have \(d^m\cdot d^m=d^{m+n}\text{.}\) With \(m=9\) and \(n=7\) we get
By the definition of powers \((a\cdot b)^n\) is the product of \(m\) copies of \(a\cdot b\text{.}\) Because of the commutative property of multiplication (see Example 1.40) we can reorder the product of \(m\) copies of \(a\cdot b\) copies such that we have the product of \(m\) copies of \(a\) times the product of \(m\) copies of \(b\text{.}\) Writing the latter as a power with base \(a\) times a power with base \(b\) we get
To extend our definition of exponentiation to all non-negative integer exponents, we must determine how to define the 0th power of an integer. We first consider an example.
Example1.67.What is \((-6)^0\) ?
We try to find out what \((-6)^0\) should be. Our definition of \((-6)^0\) should be consistent with the properties of exponentiation in Theorem 1.59. In particular Theorem 1.59 which states that for all natural numbers \(a\) and \(c\) we have
That is we want \((-6)^0\) multiplied by \((-6)^c\) to be equal to \((-6)^c\text{.}\) The only number by which we can multiply a (non-zero) number and get the number as a result is \(1\text{.}\) So for our equation to be true we must set
The argument in Example 1.67 holds not only for \((-6)\text{,}\) but for all integers (except for 0). Let \(b\) be an integer. To extend our definition of exponentiation to all non-negative integer exponents, we must determine how to define \(b^0\text{.}\) Let \(n\) be a positive integer. If we want the property in Theorem 1.59 to include the possibility of an exponent of zero, we must have \(b^0 \cdot b^n = b^{0+n} = b^n\text{.}\) If \(b\ne 0\text{,}\) the only choice for \(b^0\) that works is \(b^0 = 1\text{.}\)
When the base is \(0\text{,}\) there are multiple possibilities for \(b^0\) that would keep the properties in Theorem 1.59 correct. One possibility is defining \(0^0 := 1\text{.}\) As it does not break anything, that it does not build a contradiction into the system of mathematics, and it matches what we have found for non-zero bases, we go with this choice.
Definition1.68.
For all integers \(b\) we set \(b^0:=1\text{.}\)
We remark that some authors leave \(0^0\) undefined, while with our definition we have \(0^0=1\text{.}\)
We end out discussion of exponentiation with a table of powers (Figure 1.70).
\(b^n\)
0
1
2
3
4
5
6
7
8
9
0
1
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
2
1
2
4
8
16
32
64
128
256
512
3
1
3
9
27
81
243
729
2 187
6 561
19 683
4
1
4
16
64
256
1 024
4 096
16 384
65 536
262 144
5
1
5
25
125
625
3 125
15 625
78 125
390 625
1 953 125
6
1
6
36
216
1 296
7 776
46 656
279 936
1 679 616
10 077 696
7
1
7
49
343
2 401
16 807
117 649
823 543
5 764 801
40 353 607
8
1
8
64
512
4 096
32 768
262 144
2 097 152
16777216
13 421 7728
9
1
9
81
729
6 561
59 049
531 441
4 782 969
43 046 721
387 420 489
10
1
10
100
1 000
10 000
100 000
1 000 000
10 000 000
100 000 000
1 000 000 000
Figure1.70.Powers of integers. The rows contain the base \(b\) for \(0\le b\le 10\) and the columns contain the exponent \(n\) for \(0\le n\le 9\text{.}\)
SubsectionSquare Roots
Definition1.71.
Let \(b\) be a non-negative integer. By the square root of \(b\text{,}\) written as \(\sqrt{b}\text{,}\) we mean the non-negative number \(a\) such that \(a^2=b\text{.}\)
Some, but not all, square roots are integers. If the square root of \(b\) is an integer, we call \(b\) a perfect square.
Example1.72.Square roots of small perfect squares.
Some examples of perfect squares are \(1 = 1^2\text{,}\)\(4 = 2^2\text{,}\)\(9 = 3^2\text{,}\) and \(16 = 4^2\text{.}\) Their square roots are integers: \(\sqrt{1} = 1\text{,}\)\(\sqrt{4} = 2\text{,}\)\(\sqrt{9} = 3\text{,}\) and \(\sqrt{16} = 4\text{.}\)
If a number is given in a convenient form, it is easy to find its square root.
Example1.73.Square roots of perfect squares.
We give some more square roots of perfect squares.