Numbers Raised to the Zeroth Power

Young Vaughn, a Greek scholar in training, is in the middle of his daily lesson. Vaughn has learned all about exponents, but he wants to know what happens when you raise a number to the zeroth power. His Sage has just told him that any non-zero number raised to the zeroth power equals one. But Vaughn is a man of logic, a scholar! He won't accept this explanation without proof. Let's help Vaughn use patterns, logic, and laws of exponents to prove the Sage's advice about numbers raised to the zeroth power. First, let's look at the exponential patterns of 2. 2 to the first power is 2, 2 to the second power is 4, 2 to the third power is 8, and 2 to the fourth power is 16. Do you notice any patterns in the bottom of our table? As our exponents increase from left to right the value increases by a multiple of two. That makes sense, because 2 is the base, and the exponent tells us how many times the number 2 is multiplied. But what happens when we move from right to left? Notice that we're going in the opposite direction of the multiplication. Therefore, we divide by two each time. So, let's continue our pattern to the left. 2 divided by 2 is 1, 1 divided by 2 is 1/2, 1/2 divided by 2 is 1/4. Now let's fill in the exponential notation where the exponent decreases by one when moving from right to left. That gives us 2 to the zeroth power 2 to the power of negative one and 2 to the power of negative 2. Both patterns do confirm that 2 to the zeroth power is 1. So far the Sage appears to be right. This was just a pattern but we can use a law of exponents to write our proof. Recall the quotient of powers law states that when you divide two powers... with the same base you subtract the exponents. For example, to simplify 5 to the 7th divided by 5 to the third we subtract the exponents, to arrive at 5 to the fourth. But what about this expression, 5 to the third divided by 5 to the third? Well, we know that any number divided by itself equals one. We can also apply the quotient of powers law and subtract the 3 from 3, this expression simplifies to 5 to the zeroth power. Therefore, we can see that 5 to the zeroth power equals one. So let's use the variable 'x' to represent any base except zero according to the definition of the zeroth power rule. Then let the variable 'm' represent any exponent. Using the same logic as before, we can see that any number divided by itself equals one. And applying the quotient of powers law and simplifying shows us that any non-zero number raised to the zeroth power equals one. So, the sage is still right, and we can believe him with our proof of the zeroth power. We can apply what we know about the zeroth power to all sorts of expressions, including ones that use the laws of exponents. Let's take a look at the expression: the quantity 'y' to the 6th power, all raised to the zeroth power. Which law of exponents do you recognize here? Using the law of powers of powers, we can multiply together the inner and outer exponents. We can see that gives us 'y' raised to the zeroth power is one. Notice that the zeroth power rule holds true. Any non-zero number raised to the zeroth power is one. Okay, what about five times 'x' squared times 'y' cubed, all raised to the zeroth power. We could expand it using the powers of products law then we multiply the exponents applying the powers of powers law simplify and finally notice that each term reduces to one using the zeroth power law. Giving us a result of one. Or we could recall that any non-zero number raised to the zeroth power is one. Therefore, identifying when the zeroth power law can be used will help you make dealing with exponents quicker. Let's review numbers raised to the zeroth power and applying it to other power laws. What is the zeroth power law? Any non-zero number raised to the zeroth power equals one. This law works no matter which law of exponents you're dealing with such as powers of powers of products or quotient of powers. Vaughn needed to use logical reasoning to prove the zeroth power law of exponents, and so did you! Vaughn continues his studies and becomes a great philosopher, thanks to his skepticism and use of logic and reasoning. People come from far and wide to hear him speak. Vaughn's definitely going down in history as one of the greats.