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.
- Rules of Exponents (-1) Raised to an Exponent (07:57 mins) | Transcript
- Negative Next to a Number vs Negative in Parentheses (08:22 mins) | Transcript
(-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:- \(({-}1)^{5}=\ ?\) (Solution
- \(({-}1)^{4}=\ ?\) (Solution
- \(({-}1)^{105}=\ ?\) (Solution
- \(({-}1)^{236}=\ ?\) (Solution
- \(({-}5)^{3}=\ ?\) (Video Solution
- \(({-}7)^{4}=\ ?\) (Solution
- \((-{\text{a}})^{6}=\ ?\) (Video Solution
- \(-1^{2}=\ ?\) (Solution
- \(-2^{4}=\ ?\) (Solution
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