Hello, I'm Daniel Egger. Welcome to the Data Science Math Skills Section on Exponents and Logarithms. We'll be beginning now with the basic integer exponents. And we'll start with positive integer exponents. Positive Integer Exponents are numbers used for counting, like 1, 2, 3, and so on. When we have a number like 9, which is equal to 3 times 3. Three is a factor of nine, and it repeats, it occurs twice. In the number 27, the factor 3 occurs 3 times. And of course in the number 81, the factor 3 occurs 4 times. We can use positive integer exponents, which are numbers placed in the upper right corner of a factor to count how many times the same factor repeats in a number. So 9 equals 3 times 3 equals 3 To the 2nd, where 2 is the exponent. 27 equals 3 times 3 equals 3 to the 3rd, where 3 is the exponent. And 81 equals 3 times 3 times 3, equals 3 to the 4, where 4 is the exponent and so on. So we read this as 3 raised to the 4th power or simply 3 to the 4th for short. So let's look at another example. The number 248,832 is just equal to 12, times 12, times 12, times 12, times 12. And you can see the great convenience in being able to write this as 12 raised to the 5th power, where 5 is the exponent and this number is 12 to the 5th. Note that for historical reasons, numbers raise to the second power have special names as well as a regular name. So this is 4 to the 2nd but it's also 4 squared. And this is 4 to the 3rd, but it's also 4 cubed. Next we're going to talk about 0 as an exponent. These problems are easy, why do I say that? Because by the definition of exponents any number except 0 raised to the 0 with power equals 1. So for example, 3 to the 0 = 1, 2 to the 0 = 1, 2pi to the 0 = 1, 1 / x cubed to the 0 = 1 so long as x does not equal 0, and so on. Even for mathematicians zero to the zero a little too weird and so that is simply undefined. Next we'll look at negative integer exponents. 2 raised to the exponent -1 is read 2 to the -1. And it is equal 1 over 2 to the 1 or 1/2. 2 to the minus 2 is equal to 1 over 2 to the 2, or one quarter. And 2 to the minus 3 is equal to 1 over 2 to the 3, or one eighth. So you should see the pattern here. Raising a number to a negative exponent is the same as dividing it by the same integer if that integer were positive. Now let's consider dividing by a negative exponent. By the exact same logic, 1 over 2 to the -1, simply = 2 to the 1 or 2. 1/2 to the -2 simply = 2 to the 2nd or 2 squared or 4. 1/2 to the -3 simply = 2 to the 3rd or 2 cubed or 8. We express the general rule for negative exponents using letters to stand for just about any number. And the general rule looks like this, x to the minus n equals 1 over x to the n and 1 over x to the minus n equals x to the n. One of the most common uses of exponents is for something that we call scientific notation. Scientific notation is a way to write numbers that otherwise would have a very large number of zeros in them. What we do is we take the part of the number that has significant digits, here 5,972 and we put one digit to the left of the decimal place. So we have 5.972 and then we multiply by 10 raise to the appropriate exponent. Which in this case is 24. And 10 to the 24 means that you are going to have, you're going to move the decimal place 24 spaces to the right. Get it? So let's try another one. If we have a number less than one, we are going to be moving to decimal place to the left. And we are going to have a negative exponent. So in this case, we have the mass of an electron, and this would be 9.109 times10 to the -31 because we're moving the decimal place 31 places to the left. Key thing to remember is that we only need to keep these significant non-zero digits and we always have one digit, to the left of the decimal place. One digit to the left. And that's all you need to know for scientific notation. And that concludes our first video on exponents.