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Rules of Exponents
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Introduction

In this lesson, you will learn some of the rules of exponents and how to apply them together.


These videos illustrate the lesson material below. Watching the videos is optional.


Rules of Exponents Vocabulary


Review these math vocabulary words to help you understand this lesson better:

  • Base: the number or variable that is being multiplied by itself.
  • Power: the number of the exponent, meaning how many times the base is multiplied by itself.
  • Product: the answer when you multiply a number/variable by another number/variable.
  • Quotient: the answer when you divide a number/variable by another number/variable.

When working with exponents, it is helpful to learn the rules that can be applied to solve or simplify exponents.

The Product Rule of Exponents


The product rule states that if the bases are the same, then you can add the exponents. Notice the qualification for the product rule is the bases must be the same or you cannot add the exponents.

Product Rule of Exponents:axay=axy

Remember that you can multiply in any order, so (a)(b)=(b)(a). This means that if there are multiple bases, you can rearrange the order and add the exponents of any of the bases that are the same.

Example 1
Simplify x3x4.

This example involves variables, but it is performed in the same way as before.

\begin{align*}
&x^{3}x^{4}&\color{red}\small\text{Simplify this expression}\\&x^{3 + 4}&\color{red}\small\text{Product rule of exponents}\\&x^7 &\color{red}\small\text{Add the exponents with common base}\\ \end{align*

The Quotient Rule of Exponents


The quotient rule states that when dividing terms with the same base, you can subtract the exponents.

Remember, x4÷x3 can also be written as x4x3. Using the quotient rule of exponents, this is the same as x43 which is equal to x1, or x.

Quotient Rule of Exponents:axay=axy

Example 2
m6m2

m6m2Simplify this expressionm62Quotient rule of exponentsm4Subtract the exponents

Negative Exponents

When using the quotient rule of exponents, you will sometimes have a negative exponent in your answer. a2a5=a(25)=a3. What does this mean? What does a to the power of -3 mean? Doing this by hand looks like this:

a2a5=aaaaaaa=1a3

In other words, 1a3 is the simplified answer where there is no negative exponent.

So, any base raised to a negative exponent is actually equal to 1 over that number to the positive exponent, but in the denominator: a3=1a3.

This is also true when a base has a negative exponent in the denominator. In this case, the base is raised to the positive exponent and moved to the numerator, with 1 as the denominator.

Negative Exponents Rule: xa=1xa1xa=xa1

Example 3

x2x4

x2x4Simplify this expressionx(24)Quotient rule of exponentsx2Subtract exponents1x2Negative exponent rule

x2 can also be expressed as 1x2.

The Power Rule of Exponents


The power rule of exponents states that when you have an exponential expression, or in other words, a base raised to a power, and then that whole exponential expression is raised to another power, it is equal to the base raised to both powers multiplied together.

Power Rule of Exponents: (ab)c\nightarrowab×c\nightarrowabc

Example 4

(34)2

(34)2Simplify this expression34×2Power rule of exponents38Multiply exponents

Exponents of 1

Any base, like a, raised to the power of 1, a1, is just equal to itself, a. So a1=a.

Example 5

a4a3=a43=a1

Because a1=a, the answer is a.

Another way to simplify the expression a4a3 is by canceling like bases from the numerator and denominator like this:

a4a3=aaaaaaa=a

So again, the answer is a1, or a. Anything raised to the one power is just itself.

Exponents of 0

Any base, like a, raised to the power of 0, a0, is equal to 1. So a0=1.

Example 6

a2a2=a22=a0

To understand what this means, simplify the same expression by canceling the common bases.

a2a2=aaaa=11=1

a2a2=a0=1. This is true for any value for a, except for zero, 001. Base 0, to the power of 0 is undefined.

Applying Exponents Together

 
In some of the exercises you will do, there will be multiple steps to simplifying the expression and you may need to use multiple rules together to simplify expressions.

    Example 7
    This example has several different variables. All the variables in this expression are being multiplied together and all are being raised to an exponent.
    (m2x3ymx)2

    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 m2 and m1, as well as x3 and x1.

    According to the product rule, you should add the exponents of like terms together.

    (m2x3ymx)2Simplify the parentheses first(m2x3y1m1x1)2Note: Variables to the power of 1(m2m1x3x1y1)2Rearrange the like bases to be together(m(2+1)x(3+1)y1)2Product rule of exponents(m3x4y)2Add the exponents of likes basesm32x42y12Power rule of exponentsm6x8y2Multiply exponents and simplify

    By following the order of operations and the rules of exponents, you were able to simplify this equation down to m6x8y2.

    Example 8
    x2m3x3m2x7

    This equation has two different variables, or terms with different bases: x and m.

    x2m3x3m2x7Simplify this expressionm3x2x3m2x7Rearrange into like basesm3x(2+3)m2x7Product rule of exponentsm3x5m2x7Add the exponentsm(32)x(57)Quotient rule of exponentsmx2Subtract exponentsm1x2Negative exponent rulemx2Simplify

    Remember, m1=m=m1. The final answer is mx2.


    Things to Remember


    • The Order of Operations is PEMDAS. 
    • Product Rule: When multiplying two exponents and the bases are the same, add the powers.

    xaxb=xa+b

    • Quotient Rule: When dividing exponents and the bases are the same, subtract the powers.

    axay=axy

    • 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.

    xa=1xa

    • Power Rule: When an exponent is linked to the outside of a parentheses, multiply the powers. 

    (xa)b=x(a)(b)

    • Exponents of 0 and 1: Anything raised to the power of 0 is 1, anything raised to the power of 1 is itself.

    a0=1a1=a


    Practice Problems

    Simplify and evaluate the following expressions:
    1. Extra \left or missing \right (
      Solution
      x
      Solution: a9b8
      Details:
      In this example, the first thing you need to do is look at the (ab)4 part.

      (ab)4 is the same as ab multiplied together 4 times.

      This image shows two expressions. The top expression is a to the fifth power, b to the third power, left parenthesis a b right parenthesis to the fourth power, b. The bottom expression is expanded from the section of the top expression within parentheses, and it is left parenthesis a b right parenthesis, left parenthesis a b right parenthesis, left parenthesis a b right parenthesis, left parenthesis a b right parenthesis. 

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

      This image shows two expressions. The top expression is a to the fifth power, b to the third power, left parenthesis a b right parenthesis to the fourth power, b. The bottom expression is expanded from the section of the top expression within parentheses, and it is left parenthesis a a a a b b b b right parenthesis. 

      Thus, the part (ab)4 is the same as a4b4 and you can substitute that back into the original expression.

      This image shows two expressions. The top expression is a to the fifth power, b to the third power, left parenthesis a b right parenthesis to the fourth power, b. The bottom expression shows a rewrite of the first expression by substituting the section within parentheses. The second expression is: 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.

      a5 and a4 become a(5+4)=a9

      b3, b4 and b become b(3+4+1)=b8

      This image shows 4 expressions, each expression is a stage in simplifying to eventually reach the solution. The first expression is a to the fifth power, b to the third power, a to the fourth power, b to the fourth power, b. The second expression is a to the fifth power, a to the fourth power, b to the third power, b to the fourth power, b. The third expression is a to the 5 plus 4 power, b to the 3 + 4 + 1 power. The fourth expression is the final solution, which is a to the ninth power, b to the eighth power. 

      The final solution is: a9b8.
      )
    2. Extra \left or missing \right (
      Solution
      x
      Solution: x4y4
      )
    3. x5y3x2x6y2= (
      Video Solution
      x
      Solution: xy
      Details:

      | Transcript)
    4. m3x7m3x2= (
      Solution
      x
      Solution: x5
      Details:
      In this case, you are using the quotient rule on both variables m and x.

      m3x7m3x2

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

      m3m3=m(33)=m0=1

      This image shows the following equation: fraction m to the third power x to the seventh power over m to the third power x to the second power, equals m to the power of left parenthesis three minus three right parenthesis times the fraction x to the seventh power over x to the second power. Note: throughout the equation the variable m is blue and the variable x is green, and the dot between the terms in the equation means multiplication. 

      Note: The dot between the terms in the equation means multiplication.

      Since m(33)=m0=1 and 1 multiplied to anything is itself, you only have the x variables left.

      m(33)x7x2=m0x7x2=1x7x2

      Now look at the x variables and their exponents.

      x7x2=x(72)=x5

      1x7x2=1x(72)=1x5=x5

      So you have 1x5=x5.
      )
    5. Extra \left or missing \right (
      Video Solution
      x
      Solution: b10x7y2
      Details:

      | Transcript)
    6. Extra \left or missing \right (
      Solution
      x
      Solution: m4b2x3
      )
    7. Extra \left or missing \right (
      Solution
      x
      Solution: 108a2b4
      )