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Rules of Exponents: Applying Them Together
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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:
  1. \({\text{a}}^{5}\,{\text{b}}^{3}\left ( {\text{a}}\,{\text{b}} \right )^{4}\,{\text{b}} =\) (
    Solution
    x
    Solution: \({\text{a}}^{9}\,{\text{b}}^{8}\)
    Details:
    In this example, the first thing you need to do is look at the \(\left ({\text{a}}{\text{b}} \right )^{4}\) part.

    \(\left ({\text{a}}{\text{b}} \right )^{4}\) is the same as \(\text{ab}\) multiplied together 4 times.

    This is an image with two expressions. Top expression is a to the fifth power b to the third power left parentheses a b right parentheses to the fourth power b. The bottom expression is expanded into left parentheses a b right parentheses left parentheses a b right parentheses left parentheses a b right parentheses left parentheses a b right parentheses.

    This is the same as the variable a multiplied together 4 times and the variable b multiplied together 4 times.

    This is the same image as before but with one more third expression combining the second expression. Top expression is a to the fifth power b to the third power left parentheses a b right parentheses to the fourth power b. The bottom expression is expanded into left parentheses a b right parentheses left parentheses a b right parentheses left parentheses a b right parentheses left parentheses a b right parentheses. Third expression is left parentheses a a a a b b b b right parentheses.

    Thus, the part \(\left ({\text{a}}{\text{b}} \right )^{4}\) is the same as \({\text{a}}^{4}\,{\text{b}}^{4}\) and you can substitute that back into the original expression.

    This image has the same first expression as before, a to the fifth power b to the third power left parentheses a b right parentheses to the fourth power b. Now with an arrow pointing from the left parentheses a b right parentheses to the fourth power to a to the fourth power b to the fourth power in a second expression of a to the fifth power b to the third power a to the fourth power b to the fourth power b.

    Now you can add the exponents of factors with the same base. Remember that anything that doesn’t have an exponent actually has an invisible exponent of 1. So the variable b on the end has an exponent of 1 and you need to include that in the solution.

    \({\text{a}}^{5}\) and \({\text{a}}^{4}\) become \({\text{a}}^{\left ( 5+4 \right )}={\text{a}}^{9}\)

    \({\text{b}}^{3}\) and \({\text{b}}^{4}\) become \({\text{b}}^{\left ( 3+4+1 \right )}={\text{b}}^{8}\)

    This image has the same first expression as the previous second expression, a to the fifth power b to the third power a to the fourth power b to the fourth power b. Now with three more expressions below it in a progressive simplification. Second expression combines like variables into a to the fifth power a to the fourth power b to the third power b to the fourth power b. Third expression is a to the power of left parentheses 5 plus 4 right parentheses b to the power of left parentheses 3 plus 4 plus 1 right parentheses. Last expression is a to the ninth power b to the eighth power.

    The final solution is: \({\text{a}}^{9}\,{\text{b}}^{8}\).
    )
  2. \(\left ({\text{x}}\,{\text{y}} \right )^{3}{\text{x}}\,{\text{y}} =\) (
    Solution
    x
    Solution: \({\text{x}}^{4}\,{\text{y}}^{4}\)
    )
  3. \(\dfrac{{\text{x}}^{5}\,{\text{y}}^{3}\,{\text{x}}^{2}}{{\text{x}}^{6}{\text{y}}^{2}}=\) (
    Video Solution
    x
    Solution: \({\text{x}}\,{\text{y}}\)
    Details:

    (Applying Them Together #3 (01:15 mins) | Transcript)
     | Transcript)
  4. \(\dfrac{{\text{m}}^{3}\,{\text{x}}^{7}}{{\text{m}}^{3}{\text{x}}^{2}}=\) (
    Solution
    x
    Solution: \({\text{x}}^{5}\)
    Details:
    In this case, you are using the quotient rule on both variables m and x.

    \(\dfrac{{\color{Blue} {\text{m}}}^{3}\,{\color{DarkOrange} {\text{x}}}^{7}}{{\color{Blue} {\text{m}}}^{3}{\color{DarkOrange} {\text{x}}}^{2}}\)

    First, look at the quotient rule for the m variables.

    \(\dfrac{{\text{m}}^{3}}{{\text{m}}^{3}} = {\text{m}}^{\left ( 3-3 \right )} = {\text{m}}^{0} = 1\)

    This image contains the equation: fraction m to the third power x to the seventh power over m to the third power x to the second power this equals m to the power of left parentheses 3 minus 3 right parentheses times the fraction x to the seventh power over x to the second power. The variable m is tinted blue and the variable x is tinted orange. The dot between the terms in the equation means multiplication.

    Since \({\text{m}}^{\left ( 3-3 \right )} = {\text{m}}^{0} = 1\) and 1 multiplied to anything is itself, you only have the x variables left.

    \(\displaystyle{\color{Blue} {\text{m}}}^{\left ( 3-3 \right )}\cdot\frac{{\text{x}}^{7}}{{\text{x}}^{2}} = {\color{Blue} {\text{m}}}^{0} \cdot\frac{{\text{x}}^{7}}{{\text{x}}^{2}}={\color{Blue} 1}\cdot\frac{{\text{x}}^{7}}{{\text{x}}^{2}}\)

    Now look at the x variables and their exponents.

    \(\dfrac{{\text{x}}^{7}}{{\text{x}}^{2}} = {\text{x}}^{\left ( 7-2 \right )} = {\text{x}}^{5}\)

    \(\displaystyle{\color{Blue} 1}\cdot\frac{{\color{DarkOrange} {\text{x}}}^{7}}{{\color{DarkOrange} {\text{x}}}^{2}} = {\color{Blue} 1}\cdot {\color{DarkOrange} {\text{x}}}^{\left ( 7-2 \right )} = {\color{Blue} 1}\cdot {\color{DarkOrange} {\text{x}}}^{5} = {\color{DarkOrange} {\text{x}}}^{5}\)

    So you have \(1 \cdot {\text{x}}^{5} = {\text{x}}^{5}\).
    )
  5. \(\left ( {\text{b}}^{4}\,{\text{x}}^{3}\,{\text{y}}\,{\text{b}} \right )^{2}\,{\text{x}} =\) (
    Video Solution
    x
    Solution: \({\text{b}}^{10}\,{\text{x}}^{7}\,{\text{y}}^{2}\)
    Details:

    (Applying Them Together #5 (03:08 mins) | Transcript)
     | Transcript)
  6. \(\left ( {-}{\text{m}} \right )^{3}\,{\text{b}}^{2}\,{\text{mx}}^{3} =\) (
    Solution
    x
    Solution: \(-{\text{m}}^{4}\,{\text{b}}^{2}\,{\text{x}}^{3}\)
    )
  7. \(\left ( -3 \right )^{3}\,{\text{a}}^{2}\,{\text{b}}^{4}\left ( -2 \right )^{2} =\) (
    Solution
    x
    Solution: \(-108{\text{a}}^{2}\,{\text{b}}^{4}\)
    )

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(-1) Raised to an Exponent
Introduction to Roots