2.6: Rules of Exponents
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Overview
- The Product Rule for Exponents
- The Quotient Rule for Exponents
- Zero as an Exponent
We will begin our study of the rules of exponents by recalling the definition of exponents.
If x is any real number and n is a natural number, then
xn=x⋅x⋅x⋅…⋅x⏟n factors of x
An exponent records the number of identical factors in a multiplication.
In xn,
x is the base
n is the exponent
The number represented by xn is called a power
The term xn is read as "x to the nth."
The Product Rule for Exponents
The first rule we wish to develop is the rule for multiplying two exponential quantities having the same base and natural number exponents. The following examples suggest this rule:
x2⋅x4=xx⏟2⋅xxxx⏟4=xxxxxx⏟6=x62+4=6
a⋅a2=a⏟1⋅aa⏟2=aaa⏟3=a31+2=3
If x is a real number and n and m are natural numbers,
xnxm=xn+m
To multiply two exponential quantities having the same base, add the exponents. Keep in mind that the exponential quantities being multiplied must have the same base for this rule to apply.
Sample Set A
Find the following products. All exponents are natural numbers.
x3⋅x5=x3+5=x8
a6⋅a14=a6+14=a20
y5⋅y=y5⋅y1=y5+1=y6
(x−2y)8(x−2y)5=(x−2y)8+5=(x−2y)13
x3y4≠(xy)3+4
Since the bases are not the same, the product rule does not apply.
Practice Set A
Find each product.
x2⋅x5
- Answer
-
x2+5=x7
x9⋅x4
- Answer
-
x9+4=x13
y6⋅y4
- Answer
-
y6+4=y10
c12⋅c8
- Answer
-
c12+8=c20
(x+2)3⋅(x+2)5
- Answer
-
(x+2)3+5=(x+2)8
Sample Set B
We can use the first rule of exponents (and the others that we will develop) along with the properties of real numbers.
2x3⋅7x5=2⋅7⋅x3+5=14x8
We used the commutative and associative properties of mulitplication. In practice, we use these properties "mentally" (as signiffied by the drawing of the box). We don't actually write the second step.
4y3⋅6y2=4⋅6⋅y3+2=24y5
9a2b6(8ab42b3)=9⋅8⋅2a2+1b6+4+3=144a3b13
5(a+6)2⋅3(a+6)8=5⋅3(a+6)2+8=15(a+6)10
4x3⋅12⋅y2=48x3y2
Practice Set B
Perform each multiplication in one step.
3x5⋅2x2
- Answer
-
6x7
6y3⋅3y4
- Answer
-
18y7
4a3b2⋅9a2b
- Answer
-
36a5b3
x4⋅4y2⋅2x2⋅7y6
- Answer
-
56x6y8
(x−y)3⋅4(x−y)2
- Answer
-
4(x−y)5
8x4y2xx3y5
- Answer
-
8x8y7
2aaa3(ab2a3)b6ab2
- Answer
-
12a10b5
an⋅am⋅ar
- Answer
-
an+m+r
The Quotient Rule for Exponents
The second rule we wish to develop is the rule for dividing two exponential quantities having the same base and natural number exponents.
The following examples suggest a rule for dividing two exponential quantities having the same base and natural number exponents.
x5x2=xxxxxxx=(xx)xxx(xx)=xxx=x3. Notice that 5−2=3
a8a3=aaaaaaaaaaa=(aaa)aaaaa(aaa)=aaaaa=a5. Notice that 8−3=5
If x is a real number and n and m are natural numbers,
xnxm=xn−m,x≠0.
Sample Set C
Find the following quotients. All exponents are natural numbers.
x5x2=x5−2=x3
27a3b6c23a2bc=9a3−2b6−1c2−1=9ab5c
15x□3x△=5x□−△
The bases are the same, so we subtract the exponents. Although we don’t know exactly what □−△ is, the notation □−△ indicates the subtraction.
Practice Set C
Find each quotient
y9y5
- Answer
-
y4
a7a
- Answer
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a6
(x+6)5(x+6)3
- Answer
-
(x+6)2
26x4y6z213x2y2z
- Answer
-
2x2y4z
When we make the subtraction, n−m, in the division, xnxm, there are three possibilities for the values of the exponents:
The exponent of the numerator is greater than the exponent of the denominator, that is, n>m. Thus, the exponent, n−m, is a natural number.
The exponents are the same, that is, n=m. Thus, the exponent, n−m, is zero, a whole number.
The exponent of the denominator is greater than the exponent of the numerator, that is, n<m. Thus, the exponent, n−m, is an integer.
Zero as an Exponent
In Sample Set C, the exponents of the numerators were greater than the exponents of the denominators. Let’s study the case when the exponents are the same.
When the exponents are the same, say n, the subtraction n−n produces 0.
Thus, by the second rule of exponents, xnxn=xn−n=x0
But what real number, if any, does x0 represent? Let's think for a moment about our experience with division in arithmetic. We know that any nonzero number divided by itself is one.
88=1, 4343=1, 258258=1
Since the letter x represents some nonzero real number divided by itself. Then xnxn=1.
But we have also established that if x≠0, xnxn=x0. We now have that xnxn=x0 and xnxn=1. This implies that x0=1, x≠0.
Exponents can now be natural numbers and zero. We have enlarged our collection of numbers that can be used as exponents from the collection of natural numbers to the collection of whole numbers.
If x≠0, x0=1
Any number, other than 0, raised to the power of 0, is 1. 00 has no meaning (it does not represent a number).
Sample Set D
Find each value. Assume the base is not zero.
60=1
2460=1
(2a+5)0=1
4y0=4⋅1=4
y6y6=y0=1
2x2x2=2x0=2⋅1=2
5(x+4)8(x−1)55(x+4)3(x−1)5=(x+4)8−3(x−1)5−5=(x+4)5(x−1)0=(x+4)5
Practice Set D
Find each value. Assume the base is not zero.
y7y3
- Answer
-
y7−3=y4
6x42x3
- Answer
-
3x4−3=3x
14a77a2
- Answer
-
2a7−2=2a5
26x2y54xy2
- Answer
-
132xy3
36a4b3c88ab3c6
- Answer
-
92a3c2
51(a−4)317(a−4)
- Answer
-
3(a−4)2
52a7b3(a+b)826a2b(a+b)8
- Answer
-
2a5b2
ana3
- Answer
-
an−3
14xrypzq2xryhz5
- Answer
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7yp−hzq−5
Exercises
Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers.
32⋅33
- Answer
-
35=243
52⋅54
90⋅92
- Answer
-
92=81
73⋅70
24⋅25
- Answer
-
29=512
x5x4
x2x3
- Answer
-
x5
a9a7
y5y7
- Answer
-
y12
m10m2
k8k3
- Answer
-
k11
y3y4y6
3x2⋅2x5
- Answer
-
6x7
a2a3a8
4y4⋅5y6
- Answer
-
20y10
2a3b2⋅3ab
12xy3z2⋅4x2y2z⋅3x
- Answer
-
144x4y5z3
(3ab)(2a2b)
(4x2)(8xy3)
- Answer
-
32x3y3
(2xy)(3y)(4x2y5)
(14a2b4)(12b4)
- Answer
-
18a2b8
(38)(1621x2y3)(x3y2)
8583
- Answer
-
82=64
6463
2924
- Answer
-
25=32
416413
x5x3
- Answer
-
x2
y4y3
y9y4
- Answer
-
y5
k16k13
x4x2
- Answer
-
x2
y5y2
m16m9
- Answer
-
m7
a9b6a5b2
y3w10yw5
- Answer
-
y2w5
m17n12m16n10
x5y7x3y4
- Answer
-
x2y3
15x20y24z45x19yz
e11e11
- Answer
-
e0=1
6r46r4
x0x0
- Answer
-
x0=1
a0b0c0
8a4b04a3
- Answer
-
2a
24x4y4z0w89xyw7
t2(y4)
- Answer
-
t2y4
x3(x6x3)
a4b6(a10b16a5b7)
- Answer
-
a9b15
3a2b3(14a2b52b)
(x+3y)11(2x−1)4(x+3y)3(2x−1)
- Answer
-
(x+3y)8(2x−1)3
40x5z10(z−x4)12(x+z)210z7(z−x4)5
xnxr
- Answer
-
xn+r
axbyz5z
xn⋅xn+3
- Answer
-
x2n+3
xn+3xn
xn+2x3x4xn
- Answer
-
x
Exercises for Review
What natural numbers can replace x so that the statement −5<x≤3 is true?
Use the distributive property to expand 4x(2a+3b)
- Answer
-
8ax+12bx
Express xxxyyyy(a+b)(a+b) using exponents.
Find the value of 42+32⋅23−10⋅8
- Answer
-
8
Find the value of 42+(3+2)2−123⋅5+24(32−23+42.