Introduction
In this lesson, you will learn how to apply the rules of exponents together.
This video illustrates the lesson material below. Watching the video is optional.
Applying Exponents Together
In some of the exercises you will do, there will be multiple steps to simplifying the expression, much like in the Order of Operations. Each of these rules below is a tool and all the tools can be used together to simplify expressions.
- Product Rule: When the bases are the same, you add the powers. This happens when the variables are being multiplied.
- Power Rule: Multiply the powers. This happens when an exponent is linked to the outside of a parentheses.
- Exponents of 1: Anything raised to the \(1\) is itself.
- Follow the Order of Operations (PEDMAS).
Example 1
This example has several different variables. All the variables in this expression are being multiplied together and all are being raised to an exponent.
\begin{align*} (m^2x^3ymx)^2 \end{align*}
Use the order of operations and what you know about exponents to simplify this problem.
First, according to the order of operations you need to look at what is inside of the parentheses. Once you've simplified the variables inside the parentheses as much as possible, you can move on.
The first thing to notice is that, inside the parentheses, there are terms with the same base being multiplied together. You have \(m^2\) and \(m^1\), as well as \(x^3\) and \(x^1\).
According to the product rule, mentioned at the beginning of this chapter, you should add the exponents of like terms together.
\begin{align*}&(m^2x^3ymx)^2 &\color{red}\text{Simplify the parenthesis first}\\\\ &(m^2x^3y^1m^1x^1)^2 &\color{red}\text{Note: Variables to the power of 1}\\\\ &(m^2m^1x^3x^1y^1)^2 &\color{red}\text{Rearrange the like bases to be together}\\\\ &(m^{(2+1)}x^{(3+1)}y^1)^2 &\color{red}\text{Product rule of exponents}\\\\ &(m^3x^4y)^2 &\color{red}\text{Add the exponents of likes bases}\\\\&m^{3\cdot2}x^{4 \cdot 2}y^{1 \cdot 2} &\color{red}\text{Power rule of exponents}\\\\&m^{6}x^{8}y^{2} &\color{red}\text{Simplify and multiply exponents}\\\end{align*}
By following the order of operations and the rules of exponents, you were able to simplify this equation down to \(m^6x^8y^2\).
Things to Remember
- The Order of Operations is PEMDAS.
- Product Rule: When multiplying two exponents and the bases are the same, add the powers.
\begin{align*}x^a x^b = x^{a+b}\end{align*}
- Quotient Rule: When dividing exponents and the bases are the same, subtract the powers.
\begin{align*}\frac{a^x} {a^y} =a^{x - y}\end{align*}
- Negative Exponent Rule: When an exponent has a negative power, move it to the other part of the fraction (numerator or denominator) and the power becomes positive.
\begin{align*}x^{-a} = \frac{1}{x^a}\end{align*}
- Power Rule: When an exponent is linked to the outside of a parentheses, multiply the powers.
\begin{align*}(x^a)^b = x^{(a)(b)}\end{align*}
- Exponents of 0 and 1: Anything raised to the power of 0 is 1, anything raised to the power of 1 is itself.
\begin{align*}a^{0}=1\\a^1=a\end{align*}
- (−1) Raised to an Exponent: If the exponent is even, the answer will be positive. If the exponent is odd, the answer will be negative.
\begin{align*} (-1)^x = 1, \text{when x is EVEN} \\ (-1)^x = -1, \text{when x is ODD} \end{align*}
Practice Problems
Simplify and evaluate the following expressions:- \({\text{a}}^{5}\,{\text{b}}^{3}\left ( {\text{a}}\,{\text{b}} \right )^{4}\,{\text{b}} =\) (Solution
- \(\left ({\text{x}}\,{\text{y}} \right )^{3}{\text{x}}\,{\text{y}} =\) (Solution
- \(\dfrac{{\text{x}}^{5}\,{\text{y}}^{3}\,{\text{x}}^{2}}{{\text{x}}^{6}{\text{y}}^{2}}=\) (Video Solution
- \(\dfrac{{\text{m}}^{3}\,{\text{x}}^{7}}{{\text{m}}^{3}{\text{x}}^{2}}=\) (Solution
- \(\left ( {\text{b}}^{4}\,{\text{x}}^{3}\,{\text{y}}\,{\text{b}} \right )^{2}\,{\text{x}} =\) (Video Solution
- \(\left ( {-}{\text{m}} \right )^{3}\,{\text{b}}^{2}\,{\text{mx}}^{3} =\) (Solution
- \(\left ( -3 \right )^{3}\,{\text{a}}^{2}\,{\text{b}}^{4}\left ( -2 \right )^{2} =\) (Solution
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