What are Laws of Exponents?
When you're working with exponents, there are a few different rules that you can use to simplify or rearrange them. We call these the "Laws of Exponents."
First, let's go back to the definition of what exponents mean in the first place. Exponents represent repeated multiplication, where the bottom number (the "base") is multiplied by itself the number of times indicated by the small top number (the "power" or "exponent"). For example, $5^7$ means 5 multiplied by itself seven times, $5×5×5×5×5×5×5$, and $n^4$ means n multiplied by itself 4 times, $n·n·n·n$.
Now, using that definition, we can work out the meaning of some of the more complicated exponent expressions.
The addition rule
Suppose you have numbers being added within an exponent. This works out the same as applying each of the two exponent numbers separately and multiplying the expressions: $x^{a+b}$ means the same as $x^a·x^b$.
For example, if you have $5^{7+x}$, you could use this rule to turn it into $5^7·5^x$ , which could then be simplified to $78125·5^x$.
More often, this relationship will be used to go the other direction: if you're multiplying two exponential expressions with the same base, you can use the addition rule to combine them into a single expression with the exponents being added: $x^a·x^b$ turns into $x^{a+b}$. For example, $6^2·6^x$ would become $6^{2+x}$ , and $11^3·11^{27}$ would become $11^{3+27}$, or $11^{30}$.
To demonstrate how this works, consider $m^3·m^5$. If we write out what this means using the definition of exponents, then we get $(m·m·m)·(m·m·m·m·m)$, which becomes $m·m·m·m·m·m·m·m$, or $m^8$. If we use the addition rule instead, we can just turn $m^3·m^5$ into $m^{3+5}$, which gets us to $m^8$ without having to write out all those individual m's.
WARNING! It's important to remember that this rule only gives us a relationship between addition inside the exponent and multiplication outside the exponent. It doesn't tell us anything about multiplication inside the exponent (like $x^{a·b}$)or addition outside the exponent (like $x^a + x^b$) - one of those will have a different rule later, and the other doesn’t have an exponent rule. So be careful not to get these mixed up!
The subtraction rule
The subtraction rule is pretty similar to the addition rule. Subtraction is reverse addition, and division is reverse multiplication - so this rule says that subtracting inside the exponent is the same as applying the exponents separately and dividing the expressions: $x^{a-b}$ is the same as $\frac{x^a}{x^b}$.
For example, $3^{x-4}$ would turn into $\frac{3^x}{3^4}$, or $\frac{3^x}{81}$.
Going the other direction, if you're dividing two exponential expressions with the same base, you can combine them into a single expression by subtracting the exponents: $\frac{x^a}{x^b}$ turns into $x^{a-b}$. For example, you can turn $\frac{x^{100}}{x^{17}}$ into $x^{100-17}$, or $x^{83}$.
The multiplication rule (aka the "power to a power" rule)
Remember a few paragraphs ago when I warned that the addition rule doesn't tell us anything about multiplication inside of exponents? Now we get to find out what that does! This rule says that when you have something raised to a power, and that whole thing raised to another power, such as $(x^a)^b$, that's the same thing as multiplying the two powers together: $x^{a·b}$.
For example, $(14^2)^5$ is the same as $14^{2×5}$, or $14^{10}$.
Like the other rules, this relationship can go the opposite direction as well: $x^{a·b}$ can turn into $(x^a)^b$. Furthermore, since multiplication is commutative (meaning order doesn't matter), $x^{a·b}$ is the same thing as $x^{b·a}$, so it can also turn into $(x^b)^a$ as well! For example, $8^{12·h}$ can be rewritten as $(8^{12})^h$, or as as $(8^h)^{12}$.
The zero power rule
What happens if you try to take something to the zero power? You can't multiply something by itself zero times - there's nothing to multiply! However, using the rules we just learned, we can figure this out. Suppose we have $x^0$. We know that anything minus itself is zero, so we can use that to sneakily transform zero by writing it as some number minus itself - let's use 3. So now it's $x^{3-3}$. Using the subtraction rule, it would then turn into $\frac{x^3}{x^3}$. But when we have something divided by itself, that's just 1! So the zero power rule tells us that anything* to the zero power is 1: $x^0 = 1$.
For example, $9^0=1$, $0.7^0=1$, and $5269^0=1$.
*Well, almost anything. If it's zero to the zero power, then the zero power rule doesn't apply. $0^0$ is what we call an "indeterminate form," which is a fancy way of saying that it doesn't have a specific answer. Think of it this way: if we apply the zero power rule, then we'd get 1. On the other hand, we know that zero times anything is itself, so zero to any power would have to be zero. Since $0^0$ can't be equal to 0 and 1 at the same time, the answer is "there is no right answer!" However, some calculators will still give the answer as 1, even though that’s technically incorrect.
The negative power rule
Thanks to the subtraction rule and the zero power rule, we can also define what it means to have a negative power: $x^{-a}$. Since -a is the same thing as $0 - a$, we can turn $x^{-a}$ into $x^{0-a}$, which then becomes $\frac{x^0}{x^a}$. And since the zero power rule tells us that $x^0$ is 1, that means $x^{-a}$ is equal to $\frac{1}{x^a}$. In other words, you can turn a negative power into a positive power by using the reciprocal of the fraction.
For example, $3^{-18}$ is equal to $\frac{1}{3^{18}}$, $\frac{1}{x^-5}$ is the same thing as $x^{5}$, and $\frac{1}{x^7}$ is the same thing as $x^{-7}$
WARNING! Like the zero power rule, the negative power rule can't be used if the base is zero - zero doesn't have a reciprocal!
The product to a power rule
If you have two or more things being multiplied together and then raised to a power, you can distribute the power to each of the things being multiplied: $(ab)^c = a^cb^c$.
For example, $(2x)^5$ would be $2^5x^5$, or $32x^5$.
As with some of the other rules, we can show why this works by multiplying it out by hand. Let's use the example from before of $(2x)^5$. Taking $2x$ to the fifth power means multiplying it by itself 5 times. So that's $2x·2x·2x·2x·2x$, which can then rearrange into $2·2·2·2·2·x·x·x·x·x$, giving us $2^5x^5$, just as the rule suggests.
The quotient to a power rule
Similarly, if you have two or more things being divided and then raised to a power, you can distribute the power to each of the things being divided: $(\frac{a}{b})^c= \frac{a^c}{b^c}.$ For example, $(\frac{6}{11})^{13}$ would be $\frac{6^{13}}{11^{13}}$.
The fractional exponent rule
The last rule on our list says that you have an exponent that's a fraction (aka dividing inside an exponent), then the denominator of the fraction is the same as a root: $x^{\frac{a}{b}}$ is the same thing as $\sqrt[b]{x^a}$ or $(\sqrt[b]{x})^a$.
For example, $32^{\frac{3}{5}}$ is the same thing as $\sqrt[5]{32^3}$ or $(\sqrt[5]{32})^3$, and $(\sqrt[7]{9})^{15}$ can be rewritten as $9^{\frac{15}{7}}$.
I hope these exponent rule explanations are helpful for you! Let me know in the comments what other math topics you'd be interested in learning more about.
