What Are Exponents?

Before we can get into the essential exponent laws and how to apply them, we first need to talk about exponents and why they are so important.
So what are Exponents?
Essentially exponential numbers are repeated multiplications of a constant. We can define an exponent as a small number that is placed above or beside a number that shows how many times that number is multiplied. An example of an exponent is: 5^2 = 5 * 5 We can separate this into two parts (a base and an exponent) and put it into the following formula. 5^2 = 5*5 = 25 As you can see from the above example 5 is the base number and the exponent shows how many times we need to multiply the base number . In the example 2 was the exponent which simply meant to multiply the base number 5 by itself 2 times to reach the answer 25. A more complex example would be: 2^3 = 2*2*2 Yes, the rules do not change, so 2^3 would also work out to be 8, but to prove it we still have to show the process we can break it down: 2^3 = 2* 2 * 2 or we could jump to the point where we see that we have it solved as 2^3 = 8. As you can see – there is no magic behind exponents, they simply represent how many times you need to multiply the base number to reach the final number (which is called the result).

The Product of Powers Law

The product of powers rule states that when multiplying two exponents with the same base, you have to add the exponent. Usually, you will see a number multiplied by itself repeatedly. Numbers are expressed as an exponent to save time and make it easier to read.
Here is an example:
The base is 5, and the exponent is 2, meaning you are multiplying 5 by itself 2 times. There are two different ways to express that mathematically:
5 x 5
5^2
Solving that problem expresses the rule:
That theorem is known as the multiplication rule. However, that’s not the only way to express and write an answer for a number multiplied by itself repeatedly. You can use the variable x.
x^2 = x * x
The number x is the base. The base is the same number and the exponent shows how many times the base is used in the multiplier.
When you multiply exponents with the same base, you only have to add the exponents. You have to remember that you only do this when the bases are the same. If the bases are different, you can’t use the product of powers rule.
This works with variables too.
x^2 * x^3 = x^5
Powder x has an exponent of 2, and Potassium has an exponent of 3. These are two variables with the same base, so you can use the product of powers rule.
x^2 + x^3 = x^5
You add the exponents.
Another example:
y^5 * y^8 = y^13
y^5 + y^8 = y^13
The base is y, and the exponent is 13.

The Quotient of Powers Law

We can divide powers with the same base by subtracting exponents:
If x^m and x^n are positive powers:
(x^m)/(x^n) = x^(m-n)
This means that the exponent of the denominator is subtracted from the exponent of the numerator.
It is important to remember that in the quotient of powers rule, the exponent of the denominator is subtracted because of how it is written.
If we think about powers in the same way we think about fractions, then we can think about multiplying and dividing powers as we think about multiplying and dividing fractions. We know that when we have "a over b", if we multiply by "b over 1", we simplify the fraction to "a" by multiplying by the reciprocal.
When we think of it this way, we realize that:
(x^m)/(x^n) = (x^m)/(x^n) * (x^n)/(x^n) = (x^(m+n))/(x^n) = x^(m+n – n) = x^m
We can also think of this in a similar way as we think about proper and improper fractions.
(x^m)/(x^n) = (ohm)/(nn)
= (ohm)/(nn) = 1
As long as o is greater than n, this will not equal zero.
You just cannot get rid of the fact that n cannot be equal to 0 because we cannot divide by zero, and o must be reduced by n to make it a proper fraction.
The dividend goes inside the hauser, and the divisor goes on the outside.
As such, one cannot simply remove the divisor as per the quotient of powers rule.
Using this reasoning, you realize that the exponent in the denominator is subtracted because of how it is written.
We can take a few examples to try to understand the concept better.
(x^2)/(x^3) = x^(2-3) = x^(2-3) = x^(2-3) = x^(-3) = 1/(x^3)
Let’s say we have a classic one in mathematics:
(x^1)/(x^1) = x^(1-1) = x^(0) = 1
When we divide powers with the same base, we can do this by subtracting their exponents.

The Power of a Power Law

The power of a power rule is something that’s mostly ignored in high school maths, at least until students get to advanced functions or calculus, but for those who are interested in knowing what’s really going on in exponent-land, it’s important to know that it’s simply a matter of multiplying exponents together, so that when something is exponentiated twice, you multiply the number of exponents.
The formula is a^m^n = a^(m*n), and it’s important to have the parentheses there so that the reader knows which order to exponents occur. If you didn’t have the parentheses, you’d lose track of what’s going on. If you have multiple exponents like you do here, the parentheses are necessary to clarify.
Simple examples of this would be
(4^2)^3 which equals 4^(2*3) or 4^6, so doing the exponentiation in the order that we read it left to right according to BODMAS would lead you to an answer of 4096 (also, the proper value for the parentheses).
(2^3)^2 = 2^(3 * 2) or 2^6 = 64
(3^2)^2 = 3^(2 * 2) or 3^4 = 81
(4^6)^2 = 4^(6 * 2) or 4^12 = 1679616
and so on.
Now, something interesting happens when 2 is the base instead of 4. Even though it is the same exponent as before, when changed to a smaller number the values get much larger:
(2^6)^2 = 2^(6 * 2) or 2^12 = 4096
(2^4)^2 = 2^(4 * 2) or 2^8 = 256
(2^2)^2 = 2^(2 * 2) or 2^4 = 16
It’s not quite exponential growth, but is close.
The reason that we actually use the first formula above is that it has to do with how many times you apply the exponent. I have used a for "lambda" so that we might have something like (λ^3) ^ (λ^4) = (λ^3) ^ (λ^4) = λ^(3 * 4) = λ^12 or (λ^6) ^ (λ^3) = (λ^6) ^ (λ^3) = λ^(6 * 3) = λ^18 and so forth.

The Power of a Product Law

The power of a product rule relates to the law of exponents and states that an exponent can be distributed over a product. So, for example, (ab)n would be equal to anbn.
This is the same as saying that (ab)n = anbn. The way to consider this is to think about the (a)exponent(b)exponent for a moment. If we were to distribute an (ab) to the exponent of (a) as well as the exponent of (b), then we would have an+bn. This is the same as saying (an)(bn), which is (ab)n, but it takes a moment to get there so you might just find it simpler to use the power of a product rule as described above.
The power of a product rule can be used in multiple similar situations. For example, if you had something like (ab)c, you would be able to distribute the exponent over the (ab), which makes (ab)c the same as (a)(bc).
You could have the following: ((ab)c)2, then you do (ab)2c2, which is the same as a2b2c2.

The Power of a Quotient Law

The power of a quotient rule is an exponent rule that shows: when you raise a fraction to an exponent, the exponent applies to both the numerator and the denominator. Basically, when you see a fraction raised to an exponent, the exponent applies to both pieces, the top and bottom of the fraction. This does two things; it allows you to evaluate the expression more easily, and it gives you an additional step in evaluating.
Let’s say we have something like "a" over "b" to the "n"th power. You should know that the definition of a fraction means: "a" over "b" means "a" divided by "b." The fraction bar is like a division sign, so "a" over "b" means "a" divided by "b," and, therefore, "a" over "b" to the "n"th power means "a" to the "n"th power, divided by "b" to the "n"th power.
The power rule of exponents is: when you multiply exponential expressions with the same base, you add the exponents a/b^n = a^n/b^n. So here you could apply the power rule on both "a" to the "n"th, and "b" to the "n"th, and you’ll get it; a^n/b^n.
Let’s throw in a negative coefficient, and use "k." When you raise a fraction to a power, you apply the exponent to both the numerator and denominator. So, you have a "k" and an "m." This means that "m" has an exponent of "m," and when you simplify it down it equals what you’d expect.
If you have a negative whole number, the negative still applies to the numerator to the "n"th power. So, you have "k" to the "n"th, over "m" to the "n"th.

Law of Negative Exponents

The next thing you need to know about exponents is called negative exponents. The general principle of a negative exponent is: Every negative exponent on a number represents the reciprocal of that base raised to the absolute value of the exponent. In other words, if you have an expression , a^-b means 1 divided by a raised to the b power. Here are some examples of how this works.
If you had an expression like: 2^-3 This means: 1/(2^3) This is the same as 1/(2 * 2 * 2). This is equal to 1/8 or one eighth.
Let’s take another example of this to show how it works. If, instead of 2, you were squaring a four. 4^-2 This tells you that it is 1 over: (4^2) That’s equal to: 1/(4 * 4) or 1/(16). We can recap and review these key concepts in the form of a summary table.

Law of Zero Exponents

A common basic exponent rule is the zero exponent rule. The most basic form of the rule states that a base (any non-zero base) raised to the power of zero equals one. For example, 3^0 = 1, 900^0 = 1, x^0 = 1. The standard rule also holds true for negative bases. For example, (-7)^0 = 1. However, we need to note an exception when the base is zero. The second part of the basic zero exponent rule says that zero to the power of zero can be treated as an undefined or indeterminate form (0^0). We will discuss more about this form here.
Using our previous example, the following values are equal to each other:
3^0 = 1; -7^0 = 1; 900^0 = 1; x^0 = 1
Anything raised to the power of zero will still equal one. If we apply the above example, we can verify that 3^0 = 1 and any other number raised to the power of zero, say (-7)^0 and 900^0, will also equal one as expected. Indeed, even the greatest number you can think of or imagine multiplied by a trillion and then raised to the power of zero will still equal one.

Law of Fractional Exponents

We have already learned that:
That is all fine and good, but when we start seeing exponents expressed as fractions, it can cause a little confusion. Remember that you are really being asked to do two things…convert the fraction in the exponent to a root, then raise the base to the power of the numerator. Let’s consider one of our old friends, 16, and see if we can express it as an exponent with a fraction. The 4th root of 16 is 2, so we can rewrite 16 as 2 to the 4th power, 16 = 2^4. If we want to convert that expression to its simplest exponent form, we need to divide the exponent by the index of the root. Since 4 is the index in the above example, we will divide by 4. 16 = 2^(4/4), which is also 2 to the first power or just 2. Now that we understand how to convert roots to exponents, let’s try it the other way around. These roots will help us see fractional exponents in action. Now, back to the fractional exponent method we just learned: 8^(1/3) = the 3rd root of 8 is 2, so we can replace that expression with 2. 3√8 = 2 or 8^(1/3) = 2^1
Let’s look at another one. 5√32 = 2^(5/2) or 32^(2/5) = 5√32 or 32^(1/5) = 2^(1/2) Someone noticed that 32 is a perfect square. If you find a perfect square under the radical, it’s much easer to convert it to a fraction exponent. So here’s how to use roots to easily help calculate a fractional exponent. All of the radicals in this calculation are perfect squares, so let’s convert them to an equivalent fraction exponent. 2^(3/2) * 2^(1/2) = 2^(4/2) = 2^2 = 4 This is good to know, because you will run into these patterns often. 36^(1/2) * 4^(1/2) = 6 * 2 = 12 32^(5/3) * 8^(1/3) = 2^5 * 2^(1/3) = 2^(16/3) = 16^(2/3) = the 3rd root of 16^2 = the 3rd root of 256 = approximately 6 If you see a related pattern in your problems, be sure to use it to make your problem even easier!

Common Errors of Exponent Laws

When learning about new concepts, it can be so easy to find yourself tripping up on some of the details. So what will some of those mistakes look like with exponent rules? Here are a few common ones in order to save you time in the future.

1. Confusing big numbers for small numbers

For some people, it may make more sense to simply write out a large number as it is rather than attempting to express it in a more sensible way. But when it comes to exponents, you’ll find that the rules will help you out a lot more if you follow protocol.
To clarify this rather cryptic statement, let’s say that you found the number 62 to be far too difficult to write so you wrote it as (60). When you come to another power, you could end up making the mistake of writing it like this:

(60)2 = (60)2 = 602 = 3,600.

On the other hand, when you see the original way of writing it, 62, and you tend to think of it as 6 × 10 in scientific notation, you might be inclined to try to write it as (6 × 10)2 = (6 × 10)2 = 62 = 36 × 102 = 360,000.
Clearly, neither of these is the right answer. To avoid similar mistakes, just do the exponent math in a more basic way and use the laws properly:
62 = 6 × 10 = 62 = 36 × 10^2 = 36 × 100 = 3,600.

2. Missed zeroes and signs

Don’t forget about zeroes. You should be extra careful when there are any zeros in any equations involving exponents because some of the rules, especially with negative powers, can lose the sign if you aren’t careful. For example:
3-3 = 1/33 = 1/27 = 1/2,187 = 0.0457.

3. Getting distracted

It’s easy to get distracted while doing exponent math, whether it’s with the exponent laws or just general exponent equations. Make sure to stick to the problem and follow it through to the end.
It’s also easy to lose sight of where you are in the exponent math in your head. You may have found the solution at a earlier point but then gotten distracted as you attempted to do the latter part of the math and messed everything up as a result.
The best thing you can do to avoid this is to write down all of your steps in the process so you don’t miss anything and so you can keep track of where you double checked. The best solution is to write down all of the applicable exponent laws to help you remember them as you go through the process.

Quick Reference Guide

Here’s a summary table of all the laws of exponents.
Law Formula Information
Multiplication of Factors
a^m \cdot a^n=a^(m+n) When multiplying two or more exponential terms with the same base, you have to add the exponents.
Division of Factors
a^m/a^n=a^(m-n) When dividing two or more exponential terms with the same base , you have to subtract the exponents.
Power of a Power
(a^m)^n=a^(m*n) Exponential terms are raised to another exponent. The power of a product law states that we have to multiply the exponents.
Power of a Product
(a*b)^n=a^n*b^n Every factor inside parenthesis is raised to an exponent.
Power of a Quotient
(a/b)^n=a^n/b^n The power of a quotient law states that you need to raise every factor in the division to an exponent.
Power of a Negative Base
(-a)^(n)=((-1)^n)*(a^n) A negative base is raised to an exponent. Whenever you have even an exponent, the outcome is positive, and if you have odd exponent, the outcome is negative.