As we move into this unit on polynomial expressions, equations, and functions, we will first go over the properties of exponents that will help us in working with polynomials. Some of this may be familiar to you through work you have done in previous classes.
The expression \(b^x\) is called a power because it involves repeated multiplication. The base, \(b\), is the expression being multiplied and the exponent, \(x\), is the number of times that base is being multiplied. So in an expression like \(3^5\), the base \(3\) is being multiplied by itself \(5\) (the exponent) times. Even though this is a relatively simple explanation, the conceptual understanding of it leads to the following rules.
Product of Powers
The expression \(b^x\) is called a power because it involves repeated multiplication. The base, \(b\), is the expression being multiplied and the exponent, \(x\), is the number of times that base is being multiplied. So in an expression like \(3^5\), the base \(3\) is being multiplied by itself \(5\) (the exponent) times. Even though this is a relatively simple explanation, the conceptual understanding of it leads to the following rules.
Product of Powers
If we look at something like \(x^4\cdot x^3\) as repeated multiplication, that expression really becomes \(x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x\) or \(x^7\). It would become very tedious to write out that expanded multiplication every time, and we don’t need to because we can just add exponents when we are multiplying like bases.
Product of Powers Property: \(\large{x^a\cdot x^b=x^{\left(a+b\right)}}\)
Power of a Power
When we are rewriting a power of a power, such as \(\left(x^2\right)^4\), we have repeated multiplication within another repeated multiplication. If we first think about it as \(x^2\cdot x^2\cdot x^2\cdot x^2\) and then think about each \(x^2\) as \(x\cdot x\), then we will have a simplified expression of \(x^8\). By using this thought process, we can see that we could just multiply the exponents when we are working with a power of a power.
Power of a Power Property: \(\large{\left(x^a\right)^b=x^{ab}}\)
Quotient of Powers
Let’s start with the expression \(\Large\frac{x^7}{x^2}\). Again, if we think about this as repeated multiplication, this expression becomes \(\Large{\frac{x\cdot x\cdot x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}}\). Part of the numerator \(x\cdot x\) will divide with the denominator to \(1\) and leave us with \(x\cdot x\cdot x\cdot x\cdot x\) or \(x^5.\) We could have also gotten there by subtracting the exponents from the original expression which will work when you are dividing like bases. Note that if we had started with \(\Large\frac{x^2}{x^7}\), then we would have had a simplified expression of \(\large{\frac{1}{x^5}}\).
Quotient of Powers Property: \(\Large\frac{x^a}{x^b}=x^{\left(a-b\right)}\)
If \(a<b\), you can either think about the exponent as \(b-a\) and write the power in the denominator or you can work with negative exponents which we will discuss below.
Power of Products/Quotients
Again, we will use repeated multiplication to establish these rules. Consider the following two examples:
\(\left(x^2y^3\right)^4=x^2y^3\cdot x^2y^3\cdot x^2y^3\cdot x^2y^3=x^8y^{12}\)
\(\large\left(\frac{x^2}{y^3}\right)^{\small{4}}=\frac{x^2}{y^3}\cdot\frac{x^2}{y^3}\cdot\frac{x^2}{y^3}\cdot\frac{x^2}{y^3}=\frac{x^8}{y^{12}}\)
In each of these examples, we could have bypassed the middle step by distributing the exponent to both powers and treating them as powers of powers.
\(\left(x^2y^3\right)^4=x^2y^3\cdot x^2y^3\cdot x^2y^3\cdot x^2y^3=x^8y^{12}\)
\(\large\left(\frac{x^2}{y^3}\right)^{\small{4}}=\frac{x^2}{y^3}\cdot\frac{x^2}{y^3}\cdot\frac{x^2}{y^3}\cdot\frac{x^2}{y^3}=\frac{x^8}{y^{12}}\)
In each of these examples, we could have bypassed the middle step by distributing the exponent to both powers and treating them as powers of powers.
Power of Products Property: \(\left(x^a\cdot y^b\right)^c=x^{ac}\cdot y^{bc}\)
Power of Quotients Property: \(\Large{\left(\frac{x^a}{y^b}\right)^c=\frac{x^{ac}}{y^{bc}}}\)
Zero Exponents
We know that if we multiply like bases, we can add the exponents. We saw above that \(x^4\cdot x^3=x^7\) and if we use the same approach for \(x^4\cdot x^0\) then we know that it is equivalent to \(x^4\) because \(4+0=4\). Therefore, \(x^0\) did not affect the \(x^4\) at all when multiplied, so \(x^0\) must equal \(1\). To generalize, any expression raised to an exponent of \(0\) is equal to \(1\).
Zero Exponents Property: \(x^0=1\)
Negative Exponents
There are several ways to think about negative exponents and how to deal with these. We’ll show several examples of rewriting expressions and then work to generalize.
\(\large{x^{\small{-3}}}=x^{\small{\left(0-3\right)}}=\Large{\frac{x^0}{x^3}=\frac{1}{x^3}}\)
|
\(\large{x^{\small{7}}\cdot x^{\small{-2}}=x^{\small{\left(7+\left(-2\right)\right)}}=x^{\small{\left(7-2\right)}}}=\Large\frac{x^7}{x^2}\)
|
\(\large{x^{\small{6}}=x^{\small{\left(0-\left(-6\right)\right)}}}=\Large{\frac{x^0}{x^{-6}}=\frac{1}{x^{-6}}}\)
|
From these examples, we can see that \(x^{-a}=\Large\frac{1}{x^a}\) which can be written as \(x^a=\Large\frac{1}{x^{-a}}\).
In general, we try to rewrite expressions with positive exponents as it is more helpful and efficient in most scenarios.
Here is a summary of the rules discussed above:
\(\begin{align}&\text{Product of Powers:}\ & \ \ &x^a\cdot x^b=x^{\left(a+b\right)}\\\\
&\text{Power of a Power:}\ & \ \ &\left(x^a\right)^b=x^{ab}\\\\
&\text{Quotient of Powers:}\ & \ \ &\frac{x^a}{x^b}=x^{\left(a-b\right)}\\\\
&\text{Power of Products:}\ & \ \ &\left(x^a\cdot y^b\right)^c=x^{ac}\cdot y^{bc}\\\\
&\text{Power of Quotients:}\ & \ \ &\left(\frac{x^a}{y^b}\right)^c=\frac{x^{ac}}{y^{bc}}\\\\
&\text{Zero Exponents:}\ & \ \ &x^0=1\\\\
&\text{Negative Exponents:}\ & \ \ &x^{-a}=\large{\frac{1}{x^a}}\end{align}\)
&\text{Power of a Power:}\ & \ \ &\left(x^a\right)^b=x^{ab}\\\\
&\text{Quotient of Powers:}\ & \ \ &\frac{x^a}{x^b}=x^{\left(a-b\right)}\\\\
&\text{Power of Products:}\ & \ \ &\left(x^a\cdot y^b\right)^c=x^{ac}\cdot y^{bc}\\\\
&\text{Power of Quotients:}\ & \ \ &\left(\frac{x^a}{y^b}\right)^c=\frac{x^{ac}}{y^{bc}}\\\\
&\text{Zero Exponents:}\ & \ \ &x^0=1\\\\
&\text{Negative Exponents:}\ & \ \ &x^{-a}=\large{\frac{1}{x^a}}\end{align}\)
Now we’ll use these to rewrite expressions.
Example 1: Simplify \(\large{\frac{2x^{-4}z^{12}}{6x^8y^0z^5}}\).
Example 1: Simplify \(\large{\frac{2x^{-4}z^{12}}{6x^8y^0z^5}}\).
\(\begin{align}&\large{\frac{1z^{12}}{3x^4x^8z^5}}\ & \ \ &\text{1) Rewrite} \frac{2}{6}\ \text{as}\ \frac{1}{3}, x^{\small{-4}}\ \text{as} \frac{1}{x^4},\ \text{and}\ y^0\ \text{as}\ \ 1.\\
&\large{\frac{z^7}{3x^{12}}}\ & \ \ &\text{2) Use the Product of Powers (add exponents) and Quotient of Powers (subtract exponents) properties.}\end{align}\)
&\large{\frac{z^7}{3x^{12}}}\ & \ \ &\text{2) Use the Product of Powers (add exponents) and Quotient of Powers (subtract exponents) properties.}\end{align}\)
Example 2: Simplify \(\left(5x^{-3}y^2z\right)^{\small{-2}}\).
We have a couple options here. We can either distribute the exponent in first or we could rewrite it as \(\large{\frac{1}{\left(5x^{-3}y^2z\right)^2}}\). The steps below are shown based on distributing right away.
\(\begin{align}&5^{-2}x^6y^{-4}z^{-2}\ & \ \ &\text{1) Use the Power of a Product Property.}\\&\large{\frac{x^6}{5^2y^4z^2}}\ & \ \ &\text{2) Rewrite the negative exponents as positive exponents.}\\&\large{\frac{x^6}{25y^4z^2}}\ & \ \ &\text{3) Rewrite}\ \ 5^2\ \text{as 25.}\end{align}\)
Example 3: Simplify \(\large{\left(\frac{10x^6y^{-3}z^{-2}}{8x^{-1}y^{-7}}\right)}\cdot\left(3x^3z^5\right)^{\small{-1}}\).
Quick Check
Rewrite each expression using positive exponents only.
Quick Check Solutions
Rewrite each expression using positive exponents only.
- \(\left(4x^3\right)^{\small{-2}}\cdot x^{10}\)
- \(\left(\large{\frac{20a^9b^{-4}c^0}{18a^{16}b^{-8}c^{11}}}\right)^{\small{-2}}\)
Quick Check Solutions