In this section, you will...
derive several rules of exponents
use the rules of exponents to simplify algebraic expressions
SectionExponents
In this section, we will explore algebraic expressions that have exponents. While you may have done computations using exponents in the past, in this section we will derive several rules of exponents that allow us to work with them in a more general sense.
Supplemental Videos
The main topics of this section are also presented in the following videos:
SubsectionExponents
If a factor is repeated multiple times, then the product can be written in exponential form \(x^n\text{.}\) The positive integer exponent \(n\) indicates the number of times the base \(x\) is repeated as a factor.
\begin{equation*}
x^n=x\cdot x\cdots x.
\end{equation*}
Suppose we wanted to multiply \(x^4\) and \(x^6\text{.}\) We could expand both of these using the definition, and then use the definition to recombine the factors:
\begin{equation*}
x^4\cdot x^6 =(x\cdot x\cdot x\cdot x)\cdot(x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot)=x^{10}.
\end{equation*}
What if we wanted to multiply two expressions with much larger exponents? Expanding using the definition can be cumbersome if the exponents are large. For this reason, we have useful rules to help us simplify expressions with exponents. In the example above, notice that we could obtain the same result by adding the exponents.
\begin{equation*}
x^4\cdot x^6=x^{4+6}=x^{10}
\end{equation*}
In general, this describes the product rule for exponents. In other words, when multiplying two expressions with the same base we add the exponents.
Example31
Use the product rule for exponents to simplify the following expressions.
- \(x^5\cdot x^8\)
- \(y^{85}\cdot y^{28}\)
- \(z^a\cdot z^{b+2a}\)
Solution
- \(x^5\cdot x^8=x^{5+8}=x^{13}\)
- \(y^{85}\cdot y^{28}=y^{85+28}=y^{113}\)
- \(z^a\cdot z^{b+2a}=z^{a+(b+2a)}=z^{3a+b}\)
How does this change if we instead raise a factor involving an exponent to another exponent? Consider \((x^6)^4\text{.}\) Again, we can expand this using the definition of exponents and then recombine:
\begin{equation*}
\begin{aligned}
(x^6)^4 \amp=x^6\cdot x^6\cdot x^6\cdot x^6\\
\amp=x^{6+6+6+6}\\
\amp=x^{24}
\end{aligned}
\end{equation*}
Here we have 4 factors of \(x^6\) which is equivalent to multiplying the exponents.
\begin{equation*}
(x^6)^4=x^{6\cdot 4}=x^{24}
\end{equation*}
This describes the power rule for exponents. Now we consider raising grouped products to a power. For example,
\begin{equation*}
\begin{aligned}
(x^2y^3)^4 \amp = x^2y^3\cdot x^2y^3\cdot x^2y^3\cdot x^2y^3\\
\amp = x^2\cdot x^2\cdot x^2\cdot x^2\cdot y^3\cdot y^3\cdot y^3\cdot y^3\\
\amp = x^{2+2+2+2}\cdot y^{3+3+3+3}\\
\amp=x^8y^{12}.
\end{aligned}
\end{equation*}
After expanding, we are left with four factors of the product \(x^2y^3\text{.}\) This is equivalent to raising each of the original grouped factors to the fourth power and applying the power rule.
\begin{equation*}
(x^2y^3)^4=(x^2)^4(y^3)^4=x^8y^{12}
\end{equation*}
In general, this describes the use of the power rule for a product as well as the power rule for exponents. In summary, the rules of exponents streamline the process of working with algebraic expressions and will be used extensively as we move through our study of algebra. Given any positive integers \(m\) and \(n\) where \(x\) and \(y\) are nonzero we have the rules below.
\begin{align*}
\text{Product Rule for Exponents: }\amp x^m\cdot x^n=x^{m+n}\\
\text{Quotient Rule for Exponents: }\amp \frac{x^m}{x^n}=x^{m-n}\\
\text{Power Rule for Exponents: }\amp (x^m)^n=x^{m\cdot n}\\
\text{Power Rule for a Product: }\amp (xy)^n=x^ny^n\\
\text{Power Rule for a Quotient: }\amp \left(\frac{x}{y}\right)^n=\frac{x^n}{y^n}.
\end{align*}
These rules allow us to efficiently perform operations with exponents.
Example32
Simplify \(\frac{10^4\cdot 10^{12}}{10^3}\text{.}\)
Solution
\begin{equation*}
\begin{aligned}
\frac{10^4\cdot 10^{12}}{10^3}\amp = \frac{10^{16}}{10^3} \amp \alert{\text{ Product Rule}}\\
\amp = 10^{16-3}\amp\alert{\text{ Quotient Rule}}\\
\amp=10^{13} \amp
\end{aligned}
\end{equation*}
In the previous example, notice that we did not multiply the base 10 times itself. When applying the product rule, we add the exponents and leave the base unchanged.
Example33
Simplify \(\left(x^5\cdot x^4\cdot x\right)^2\text{.}\)
Solution
Note that the variable \(x\) is assumed to have an exponent of one, \(x=x^1\text{.}\)
\begin{equation*}
\begin{aligned}
\left(x^5\cdot x^4\cdot x\right)^2\amp =\left(x^{5+4+1}\right)^2\\
\amp =(x^{10})^2\\
\amp =x^{10\cdot 2}\\
\amp = x^{20}
\end{aligned}
\end{equation*}
In the previous example, we made a choice to apply the product rule before the power rule; however, we could have reverse these steps:
\begin{equation*}
\begin{aligned}
\left(x^5\cdot x^4\cdot x\right)^2\amp =(x^5)^2\cdot(x^4)^2\cdot x^2 \\
\amp=x^{5\cdot2}\cdot x^{4\cdot2}\cdot x^2 \\
\amp= x^{10}\cdot x^8\cdot x^2 \\
\amp= x^{10+8+2} \\
\amp= x^{20}
\end{aligned}
\end{equation*}
In general, there will often be multiple different orders that you could apply the different exponent rules that are all mathematically correct. As long as you apply the rules carefully and correctly, you can choose to apply the rules in whichever order makes most sense for you.
When applying the exponent rules, the base could in fact be any algebraic expression.
Example34
Simplify \((x+y)^9(x+y)^{13}\text{.}\)
Solution
Treat the expression \((x+y)\) as the base.
\begin{equation*}
\begin{aligned}
(x+y)^9(x+y)^{13}\amp =(x+y)^{9+13}\\
\amp =(x+y)^{22}
\end{aligned}
\end{equation*}
The commutative property of multiplication allows us to to reorder factors to group them according to base. This is helpful in multiplying expressions with multiple variables.
Example35
Simplify \(2x^2y^3\cdot 4xy^5\text{.}\)
Solution
We first reorder the factors so that all of the coefficients are together, all of the \(x\)'s are together, and all of the \(y\)'s are together.
\begin{equation*}
\begin{aligned}
2x^2y^3\cdot 4xy^5 \amp= 2\cdot4\cdot x^2\cdot x\cdot y^3\cdot y^5 \\
\amp= 8 \cdot x^{2+1}\cdot y^{3+5} \\
\amp= 8x^3y^8
\end{aligned}
\end{equation*}
Example36
Simplify \(-8x^5y\cdot 3x^7y^3\text{.}\)
Solution
We first reorder the factors so that all of the coefficients are together, all of the \(x\)'s are together, and all of the \(y\)'s are together.
\begin{equation*}
\begin{aligned}
-8x^5y\cdot 3x^7y^3 \amp =(-8)\cdot 3\cdot x^5\cdot x^7\cdot y^1\cdot y^3\\
\amp =-24\cdot x^{5+7}\cdot y^{1+3}\\
\amp = -24x^{12}y^{4}
\end{aligned}
\end{equation*}
Division involves the quotient rule for exponents.
Example37
Simplify \(\frac{33x^7y^5z^{10}}{11x^6yz^3}\text{.}\)
Solution
\begin{equation*}
\begin{aligned}
\frac{33x^7y^5z^{10}}{11x^6yz^3} \amp = \frac{33}{11}\cdot x^{7-6}\cdot y^{5-1}\cdot z^{10-3}\\
\amp =3x^1y^4z^7\\
\amp =3xy^4z^7
\end{aligned}
\end{equation*}
The power rule for a quotient allows us to apply that exponent to the numerator and denominator. This rule requires that the denominator is nonzero and so we will make this assumption for the remainder of the section.
Example38
Simplify \(\left(\frac{-4a^2b}{c^4}\right)^3\text{.}\)
Solution
First apply the power rule for a quotient and then the power rule for a product.
\begin{equation*}
\begin{aligned}
\left(\frac{-4a^2b}{c^4}\right)^3 \amp = \frac{(-4a^2b)^3}{(c^4)^3}\amp \alert{\text{ Power Rule for a Quotient}}\\
\amp =\frac{(-4)^3(a^2)^3(b)^3}{(c^4)^3}\amp \alert{\text{ Power Rule for a Product}}\\
\amp =\frac{-64a^6b^3}{c^{12}} \amp
\end{aligned}
\end{equation*}
Using the quotient rule for exponents, we can define what it means to have zero as an exponent. Consider the following calculation:
\begin{equation*}
1=\frac{25}{25}=\frac{5^2}{5^2}=5^{2-2}=5^0.
\end{equation*}
Twenty-five divided by twenty-five is clearly equal to one, and when the quotient rule for exponents is applied, we see that a zero exponent results. In general, given any nonzero real number \(x\) and integer \(n\text{,}\)
\begin{equation*}
1=\frac{x^n}{x^n}=x^{n-n}=x^0.
\end{equation*}
This leads us to the definition of zero as an exponent,
\begin{equation*}
x^0=1,\text{ for }x\neq 0.
\end{equation*}
It is important to note that \(0^0\) is indeterminate. If the base is negative, then the result is still positive one. In other words, any nonzero base raised to the zero power is defined to be equal to one. In the following examples assume all variables are nonzero.
Example39
Simplify
- \((-2x)^0\) and
- \(-2x^0.\)
Solution
- Any nonzero quantity raised to the zeroth power is equal to \(1\text{.}\)\begin{equation*} (-2x)^0=1 \end{equation*}
- In this example, the base is \(x\text{,}\) not \(-2x\text{.}\)\begin{equation*} \begin{aligned} -2x^0\amp = -2\cdot x^0\\ \amp = -2\cdot 1\\ \amp = -2 \end{aligned} \end{equation*}
Noting that \(2^0=1\) we can write
\begin{equation*}
\frac{1}{2^3}=\frac{2^0}{2^3}=2^{0-3}=2^{-3}.
\end{equation*}
In general, given any nonzero real number \(x\) and integer \(n\text{,}\)
\begin{equation*}
\frac{1}{x^n}=\frac{x^0}{x^n}=x^{0-n}=x^{-n}\text{ for }x\neq 0.
\end{equation*}
This leads us to the definition of negative exponents:
\begin{equation*}
x^{-n}=\frac{1}{x^n}\text{ for }x\neq 0.
\end{equation*}
An expression is completely simplified if it does not contain any negative exponents.
Example40
Simplify \((-4x^2y)^{-2}\text{.}\)
Solution
Rewrite the entire quantity in the denominator with an exponent of \(2\) and then simplify further.
\begin{equation*}
\begin{aligned}
(-4x^2y)^{-2}\amp =\frac{1}{(-4x^2y)^2}\\
\amp =\frac{1}{(-4)^2(x^2)^2(y)^2}\\
\amp = \frac{1}{16x^4y^2}
\end{aligned}
\end{equation*}
Sometimes negative exponents appear in the denominator.
Example41
Simplify \(\frac{x^{-3}}{y^{-4}}\text{.}\)
Solution
\begin{equation*}
\frac{x^{-3}}{y^{-4}} =\frac{\frac{1}{x^3}}{\frac{1}{y^4}}=\frac{1}{x^3}\cdot\frac{y^4}{1}=\frac{y^4}{x^3}
\end{equation*}
The previous example suggests a property of quotients with negative exponents. Given any integers \(m\) and \(n\) where \(x\neq 0\) and \(y\neq 0\text{,}\) then
\begin{equation*}
\frac{x^{-n}}{y^{-m}}=\frac{\frac{1}{x^n}}{\frac{1}{y^n}}=\frac{1}{x^n}\cdot\frac{y^m}{1}=\frac{y^m}{x^n}.
\end{equation*}
This leads us to the property
\begin{equation*}
\frac{x^{-n}}{y^{-m}}=\frac{y^m}{x^n}.
\end{equation*}
In other words, negative exponents in the numerator can be written as positive exponents in the denominator and negative exponents in the denominator can be written as positive exponents in the numerator.
Example42
Simplify \(\frac{-5x^{-3}y^3}{z^{-4}}\text{.}\)
Solution
Notice the coefficient \(-5\text{:}\) recognize that this is the base and that the exponent is actually positive one: \(-5=(-5)^1\text{.}\) Hence, the rules of negative exponents do not apply to this coefficient which is why we leave it in the numerator.
\begin{equation*}
\begin{aligned}
\frac{-5x^{-3}y^3}{z^{-4}}\amp =\frac{-5 \alert{x^{-3}}y^3}{\alert{z^{-4}}}\\
\amp =\frac{-5y^3\alert{z^4}}{\alert{x^3}}
\end{aligned}
\end{equation*}
In summary, given integers \(m\) and \(n\) where \(x,y\neq 0\text{,}\) we have the rules below.
\begin{equation*}
\begin{aligned}
\text{Zero Exponent: }\amp x^0=1\\
\text{Negative Exponent: }\amp x^{-n}=\frac{1}{x^n}\\
\text{Quotients with Negative Exponents: }\amp \frac{x^{-n}}{y^{-m}}=\frac{y^m}{x^n}
\end{aligned}
\end{equation*}
Furthermore, all of the rules of exponents defined so far extend to any integer exponents. We can expand the scope of these properties to include any real number exponents.