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(-1) Raised to an Exponent
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Introduction

In this lesson, you will learn what happens when you multiply (−1) to itself multiple times. You will also learn a pattern that makes simplifying exponent problems with a negative base much easier.


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


(-1) Raised to an Exponent

Example 1
Simplify \( (-1)^{2} \)
\begin{align*} (-1)^{2} = (-1)(-1) = 1 \end{align*}

Example 2
Simplify \( (-1)^{3} \)

\begin{align*} (-1)^{3} =(-1)(-1)(-1)=-1 \end{align*}

Example 3
Simplify \( (-1)^{4} \)

\begin{align*} (-1)^{4}= (-1)(-1)(-1)(-1) =1 \end{align*}

Notice that when the exponent is even, it yields a (+1). When the exponent is odd, it yields a (-1).

\begin{align*}
\large\text{Rule for \((-1)\) Raised To An Exponent}: \end{align*}

\begin{align*} (-1)^x = 1, \text{when x is EVEN} \\ (-1)^x = -1, \text{when x is ODD} \end{align*}

You can actually apply this to any number that is negative.

Example 4
Simplify \( (-m)^{4} \).

This is equal to \( (-1)^{4}\times m^{4} \) according to the rules for multiplication, because \( (-m)= (-1)(m)\) all raised to the power of four. Now, use the rule for (-1) raised to an exponent. The exponent is an even number (4), so \( (-1)^{4}=1 \). Therefore:

\begin{align*} (-m)^{4} = (-1)^{4}\times (m)^{4} = 1\times m^{4} = m^{4} \end{align*}

Example 5
Simplify \( (-m)^{5} \)

\begin{align*} (-m)^{5} = (-1)^{5}\times (m)^{5} = (-1)\times (m)^{5} = -m^{5} \end{align*}

In review, if (-1) has an exponent that is EVEN, such as 2, 4, 6, 8, then the solution is a (+1). If (-1) is raised to an ODD exponent such as 1, 3, 5, 7, then the solution is -1.

    Negative Next to a Number vs. Negative in Parentheses

    It is important to know that \( −b^x \) is different than \( (−b)^x \). When you have \( −b^x \) , where the negative is not inside the parentheses, the exponent does not apply to it. This is because of the order of operations.

    Example 6

    \begin{align*} &-b^{2} &\color{red}\small\text{Simplify the expression}\\\\& (-1)\times b^{2}&\color{red}\small\text{The exponent only applies to \(b\)}\\\\& -b^{2}&\color{red}\small\text{Multiply by \(-1\)}\\\\ \end{align*}

    Example 7

    \begin{align*} &(-b)^{2} &\color{red}\small\text{Simplify the expression}\\\\
    & (-b)(-b) &\color{red}\small\text{Write \(-b\) twice}\\\\
    & (-1)(b)(-1)(b) &\color{red}\small\text{Factor a \(-1\) from \(-b\)}\\\\
    & (-1)(-1)(b)(b) &\color{red}\small\text{Rearrange into like-terms}\\\\
    & (-1)^{2}(b) ^{2} &\color{red}\small\text{Power rule of exponents}\\\\
    & 1\times b^2 &\color{red}\small\text{Simplify \( (-1)^{2} = 1\) }\\\\
    & b^2 &\color{red}\small\text{Multiplication property} \end{align*}

    Example 8

    \begin{align*} &-2^{3} &\color{red}\small\text{Simplify the expression}\\\\& 2^{3}\times-1&\color{red}\small\text{The exponent only applies to the \(2\)}\\\\& 8\times -1&\color{red}\small\text{Simplify \(2^3\)}\\\\ & -8&\color{red}\small\text{Multiplication property}\\\\\end{align*}

    Example 9

    \begin{align*} &(-2)^{3} &\color{red}\small\text{Simplify the expression}\\\\ & (-2)(-2)(-2) &\color{red}\small\text{Write \(-2\) three times}\\\\ & (-1)(2)(-1)(2)(-1)(2) &\color{red}\small\text{Factor a \(-1\) from \(-2\)}\\\\ & (-1)(-1)(-1)(2)(2)(2) &\color{red}\small\text{Rearrange into like-terms}\\\\ & (-1)^{3}(2) ^{3} &\color{red}\small\text{Power rule of exponents}\\\\ & -1\times 8&\color{red}\small\text{Simplify \( (-1)^{3} = -1\) and \(2^3 = 8\) }\\\\ & -8 &\color{red}\small\text{Multiplication property} \end{align*}

    Examples 8 and 9 have the same answer, -8. However, if the exponent in the beginning of each example had been an even number, there would have been different answers. See Examples 10 and 11.

    Example 10

    \begin{align*} & -2^{4} &\color{red}\small\text{Simplify the expression}\\\\ & -1 (2^4) &\color{red}\small\text{Factor a \(-1\) from\(-2\)}\\\\ & -1(16)&\color{red}\small\text{Simplify \( 2^4 = 16\) }\\\\ & -16 &\color{red}\small\text{Multiplication property} \end{align*}

    Example 11
    \begin{align*} & (-2)^{4} &\color{red}\small\text{Simplify the expression}\\\\ & (-1 \cdot 2)^{4} &\color{red}\small\text{Factor a \(-1\) from \(-2\)}\\\\ & (-1)^4 (2)^{4} &\color{red}\small\text{Power rule of exponents}\\\\ & 1(16)&\color{red}\small\text{Simplify \((-1)^4 =1\) and \( 2^4 = 16\) }\\\\ & 16 &\color{red}\small\text{Multiplication property} \end{align*}

    Notice that the answers are not the same. If the exponent is even, then the answers are actually different in sign. That’s why it’s important to know the difference between having the negative within parentheses being raised to a power and the negative not in parentheses with something being raised to a power.


    Things to Remember


    • If the power is even with a negative base,  then the answer is positive.
    • If the power is odd with a negative base, then the answer is negative.
    • A negative times a negative is equal to a positive.

    Practice Problems

    Evaluate the following expressions:
    1. \(({-}1)^{5}=\ ?\) (
      Solution
      x
      Solution: \(-1\)
      Details:
      The rule is that \((−1)\) raised to an odd-numbered power is negative.

      Since 5 is an odd number, the answer is \(−1\).

      The following may help illustrate why this is true.

      \((-1)^{5}\) is the same as \((−1)\) multiplied together 5 times.

      This image shows ( negative 1 highlighted in blue)(negative 1 highlighted in purple)(negative 1 highlighted in yellow)(negative 1 highlighted in green)(negative 1 highlighted in black). The parentheses next to each other indicate multiplication.

      Apply the following rules of multiplication:
      • A negative times a negative is a positive number
      • A negative times a positive is a negative number
      Take the first and second \((−1)\) and multiply them together to get 1. Do the same thing for the third and fourth \((−1)\).

      This image shows (negative 1 highlighted in blue)(negative 1 highlighted in purple)(negative 1 highlighted in yellow)(negative 1 highlighted in green)(negative 1 highlighted in black). By grouping the first two negative ones together, we have a result of positive one list below the negative one, the one that is highlighted in purple. By grouping the middle two negative ones, we have a result of positive one list below negative one, the one that's highlighted in yellow. Therefore, we have (positive one) (positive one) (negative one) listed at the bottom of this image. The parentheses next to each other indicate multiplication.

      This leaves you with \((1)(1)(−1)\). Since 1 multiplied to anything is just that thing, all you have left is \((−1)\).

      This image shows (positive one) (positive one) (negative one) equal to (negative one). The parentheses next to each other indicate multiplication.
      )
    2. \(({-}1)^{4}=\ ?\) (
      Solution
      x
      Solution: 1
      )
    3. \(({-}1)^{105}=\ ?\) (
      Solution
      x
      Solution: \(-1\)
      )
    4. \(({-}1)^{236}=\ ?\) (
      Solution
      x
      Solution: 1
      )
    5. \(({-}5)^{3}=\ ?\) (
      Video Solution
      x
      | Transcript)
    6. \(({-}7)^{4}=\ ?\) (
      Solution
      x
      Solution: 2401
      )
    7. \((-{\text{a}})^{6}=\ ?\) (
      Video Solution
      x
      Solution: \({\text{a}}^{6}\)
      Details:

      ((-1) Raised to an Exponent #7 (01:12 mins) | Transcript)
      | Transcript)
    8. \(-1^{2}=\ ?\) (
      Solution
      x
      Solution: \(-1\)
      Details:
      In the previous examples, the negative was inside the parentheses, so it was being raised to the exponent as well. If there aren’t parentheses, the exponent doesn’t apply to it. This is due to the order of operations.

      \(-1^{2} = -(1)(1) = -(1) = -1\)
      )
    9. \(-2^{4}=\ ?\) (
      Solution
      x
      Solution: \(-16\)
      Details:
      Since the negative is not inside the parentheses, the exponent doesn’t apply to it. A negative is also the same as multiplying by \(−1\).

      \(-2^{4} = (-1)(2 \times 2 \times 2 \times 2) = (-1)(16) = -16\)
      )

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