Powers of negative numbers

What is $(-1)^{100}$? Or $(-2)^{57}$? When the base is neg­a­tive, the answer can be pos­i­tive or neg­a­tive, and the expo­nent decides which. In this les­son you will learn laws 9 and 10, which tell you the sign of any power of a neg­a­tive num­ber at a glance.

Law 9: pow­ers of minus one#

Mul­ti­ply $-1$ by itself a few times. Remem­ber: a minus times a minus is a plus.

$$\begin{aligned} (-1)^2 &= (-1) \times (-1) = 1 \\ (-1)^3 &= (-1) \times (-1) \times (-1) = -1 \\ (-1)^4 &= (-1) \times (-1) \times (-1) \times (-1) = 1 \end{aligned}$$

The minus signs can­cel in pairs. With an even num­ber of them, all are paired up and the answer is $1$. With an odd num­ber, one is left over and the answer is $-1$.

Law 9. $(-1)^{\text{odd number}} = -1$ and $(-1)^{\text{even number}} = 1$.

Exam­ple 1#

$$(-1)^{100} = 1 \qquad (-1)^{201} = -1 \qquad (-1)^{48} = 1 \qquad (-1)^{15} = -1$$

$$(-1)^{1027} = -1 \qquad (-1)^{992} = 1 \qquad (-1)^{20001} = -1$$

To decide, look only at the last digit of the expo­nent. If it is $0$, $2$, $4$, $6$ or $8$, the expo­nent is even.

Exam­ple 2#

Find $(-1)^2 + (-1)^3 + (-1)^5$.

$$\begin{aligned} (-1)^2 + (-1)^3 + (-1)^5 &= 1 + (-1) + (-1) \\ &= 1 - 1 - 1 \\ &= -1 \end{aligned}$$

Exam­ple 3: neg­a­tive expo­nents#

$$\displaystyle (-1)^{-2} = \frac{1}{(-1)^2} = \frac{1}{1} = 1 \qquad (-1)^{-5} = \frac{1}{(-1)^5} = \frac{1}{-1} = -1$$

Law 9 still holds: $-2$ is even and $-5$ is odd.

Law 10: pow­ers of any neg­a­tive num­ber#

Now try $-2$.

$$\begin{aligned} (-2)^3 &= (-2) \times (-2) \times (-2) = -8 = -2^3 \\ (-2)^4 &= (-2) \times (-2) \times (-2) \times (-2) = 16 = 2^4 \end{aligned}$$

The num­ber part is the same as for $2$. Only the sign changes, and it fol­lows the same rule as for $-1$.

Law 10. $(-a)^n = -a^n$ if $n$ is odd, and $(-a)^n = a^n$ if $n$ is even.

Here $-a^n$ means "the neg­a­tive of $a^n$". For exam­ple, $-2^3 = -(2^3) = -8$.

Exam­ple 4#

$$(-2)^{57} = -2^{57} \qquad (-2)^{100} = 2^{100} \qquad (-7)^3 = -7^3 \qquad (-9)^{224} = 9^{224}$$

Exam­ple 5: find­ing val­ues#

$$\begin{aligned} (-3)^3 &= -3^3 = -27 \\ (-3)^4 &= 3^4 = 81 \\ (-4)^5 &= -4^5 = -1024 \\ (-2)^5 &= -2^5 = -32 \\ (-5)^4 &= 5^4 = 625 \\ (-15)^2 &= 15^2 = 225 \\ (-8)^3 &= -8^3 = -512 \end{aligned}$$

Exam­ple 6#

Find $(-4)^2$ and $(-4)^3$.

$$(-4)^2 = 4^2 = 16 \qquad (-4)^3 = -4^3 = -64$$

The cube has an odd expo­nent, so the answer must be neg­a­tive. Writ­ing $64$ here is a com­mon slip.

Exam­ple 7#

Find $(-2)^6 + (-4)^3$.

$$(-2)^6 + (-4)^3 = 2^6 - 4^3 = 64 - 64 = 0$$

Exam­ple 8: a neg­a­tive base with a neg­a­tive expo­nent#

$$\displaystyle (-4)^{-2} = \frac{1}{(-4)^2} = \frac{1}{4^2} = \frac{1}{16}$$

$$\displaystyle (-9)^{-10} = \frac{1}{(-9)^{10}} = \frac{1}{9^{10}}$$

Brack­ets make a dif­fer­ence#

$(-3)^4$ and $-3^4$ look alike but are dif­fer­ent.

• $(-3)^4$ means $(-3) \times (-3) \times (-3) \times (-3) = 81$.
• $-3^4$ means $-(3 \times 3 \times 3 \times 3) = -81$.

With­out the bracket, the power belongs only to the $3$, not to the minus sign.

Remem­ber. Even expo­nent: the answer is pos­i­tive. Odd expo­nent: the answer keeps the minus sign.

Com­mon mis­takes#

• Think­ing any power of a neg­a­tive num­ber is neg­a­tive. $(-3)^4 = 81$ is pos­i­tive.
• Los­ing the minus sign for an odd expo­nent. $(-4)^3 = -64$, not $64$.
• Read­ing $-3^4$ as $(-3)^4$. With­out a bracket, $-3^4 = -81$.
• Think­ing a neg­a­tive expo­nent makes the answer neg­a­tive. $\displaystyle (-4)^{-2} = \frac{1}{16}$.

Prac­tice#

1. Find $(-1)^{109}$.
2. Find $(-1)^{222}$.
3. Find $(-1)^7 + (-1)^8$.
4. Find $(-5)^4$.
5. Find $(-3)^5$.
6. Find $(-6)^2$.
7. Find $(-10)^5$.
8. Write $(-9)^{29}$ using $9^{29}$.
9. Find $(-1)^{-8}$ and $(-1)^{-7}$.
10. Find $-3^4$ and $(-3)^4$.

Answers#