减1法则

(Video) Intro to Evaluating Exponents

by PatrickJMT

Patrick JMT has a good video example of raising $-1$ to a power.

$\large{\text{Summary}}$

Exponentiation is a mathematical operation, after all. Along with the rest of the order of operations, we have to simplify exponents. We'll focus on integer exponentiation, where the base is a negative integer.

Suppose you needed to evaluate $(-1)^7$.

This can be looked at as $-1$ multiplied $7$ times. It's an example of repeated multiplication of the base. It only really makes sense for integer exponents. $$\begin{align} (-1)^7&=(-1)(-1)(-1)(-1)(-1)(-1)(-1)\\ &=1\cdot1\cdot1(-1)\\ &=1\cdot(-1)\\ &=-1 \end{align}$$

This brings us to the rule. If $-1$ is raised to an even power, it will equal $1$. If $-1$ is raised to an odd power, it will equal $-1$. $$(-1)^a=\begin{cases} -1 & a=\text{odd} \\ 1 & a=\text{even } \end{cases}$$

We can do this for other negative numbers too. For instance,

$$\begin{align} (-2)^1 &= -2 \\ (-2)^2 &= (-2)(-2) = 4 \\ (-2)^3 &= (-2)(-2)(-2) = -8 \\ (-2)^4 &= 16 \end{align}$$

The signs alternative between negative and positive. This pattern will continue if we look for larger powers of $-2$. The only important part is that we work with negative bases. If we just want to deal with positive integers, there are other rules for that. Fractional exponents, on the other hand, are more complicated to handle. We don't deal with those here.

Powers of -1

For example: $$\begin{align} (-1)^{-1} &= -1, \text{ since $-1$ is odd}\\ (-1)^0 &= 1, \text{ since $0$ is even}\\ (-1)^{53} &= -1, \text{ since $53$ is odd}\\ (-1)^{122{,}000}&= 1, \text{ since $122{,}000$ is even}\\ \end{align}$$

Here's a quick side note. We can raise negative numbers besides $-1$ to powers. The pattern is just easiest to see when $-1$ is the base.

Negatives raise normally, but every other power will also be negative

When you raise negative numbers to a power, their absolute value will increase normally, but the resulting number will be negative whenever the exponent is odd.

Think of it in terms of multiplication: when you multiply an odd number of negatives, the product is always negative. When you're raising something to the $3$rd or $5$th power, you're essentially multiplying it by itself $3$ or $5$ times.

Meanwhile, if you raise it to the $2$nd or $4$th power, you're multiplying it by itself an even number of times.

This pattern continues for larger powers too. It might be really hard to calculate $(-5)^{515}$. However, $515$ is an odd number, so we at least know $(-5)^{515}$ is negative.