The laws of exponents are rules about specific cases of multiplication. With each of these specific cases we will learn how to recognize them and how to re-write them in a way to make them easier to work with. From background knowledge we know that exponents are another form of multiplication but now we will look at some of the basic laws.

x^1 = x

This is a law that says any base raised to the 1st power is itself. One way to think of this is some base times itself once has to be itself because the base is not being changed.

x^(-1) = 1/x

This law is similar to the last because it is the same idea just working in the different direction. If the exponent represents how many times the base is multiplied by itself we can think that division is the opposite of multiplication and we represent this opposite by placing a negative sign in front of it.

x^m*x^n = x^(m+n)

This law says whenever you are multiplying like bases you can add their exponents together to get one new exponent. An example of this is 7*7*7*7*7=(7*7)*(7*7*7)=7^2*7^3=7^(2+3)=7^5. From this example we can see that the only difference is the way we write this expression. In retrospect when you originally count the power of the base you are adding 1 for each multiplication which is just a simplified version of this rule.

x^m/x^n = x^(m-n)

This law is very similar to the last and this is because we are just thinking that if we are dividing by a bas every time we divide by that it cancles out one multiplication and this can be represented by subtracting a power for every power we divide by. An example of this is 7*7*7*7/(7*7)=7^4/(7^2)=7^(4-2)=7^2.

x^0 = 1

This law is usually the most difficult for people to grasp but now that we understand some other laws of exponents this law will seem much easier to understand. The best way to explain this is with an example like the following: 7/7 =1. Since basic mathematics we know that anything divided by itself is one so let’s write this problem in a fashion that you can understand where the 0th power comes into play. 7/7=7^1/(7^1)=7^(1-1)=7^0=1 From this problem we can see that the previous law helps us find that anything divided by itself is really that number to the 0th power which in turn makes it one.

(x^m)^n = x^(m*n)

This law describes how an exponent can be raised by another exponent. A simpler way to think of this is an exponent is a group of multiplication of its bases, and in this problem the second exponent (n) is just groups of the group below it. An example of this is (7^2)^3=(7*7)^3=(7*7)(7*7)(7*7)=7^6; from this we can see that we have 3 groups of 7^2 which is 3 groups of 2 groups of 7 so to simplify it we write this expression as 6 groups of 7 or 7^6 .

(x*y)^n=x^n*y^n

This law describes the distributive property that everything that is raised to a power can be split and the exponent will carry. One example we can break down and reconstruct is 36=6^2=(3*2)^2=3^2*2^2=9*4=36 from this we can see that anything raised to a power can be properly manipulated as long as we carry our exponent.

(x/y)^n=x^n/(y^n)

This law is very similar to the last in the aspect that we can manipulate a variable as long as we distribute our exponent. An example would be (3/2)^2=3^2/(2^2)= 9/4; if we use a calculator we can see that the value of (3/2)^2 is indeed 9/4 and that validates our formula.

Now that we have a firm grasp of the laws of exponents we can refer to them as our tools for simplifying and solving various multiplication and division problems.