Howdy y'all, Mr. Cosi here coming to you from beautiful Atascosita, Texas with another algebra lesson and today we're going to talk about exponents and the laws of exponents. So get a piece of paper and a pencil and let's take some notes. In this lesson you'll need to know your math facts. You need to know how to abstract multiply. You need to know math vocabulary.
If you don't know what a sum is or a difference or product and etc you need to get that down. And you need to know your fractions. Very important skill in algebra is to know your fractions. And in this lesson you will learn what is an exponent. You'll learn how to use exponents.
And you will learn the laws of exponents. First, what's an exponent? An exponent shows how many times a number is a factor. So look at our example there.
We have x to the m power. And what that tells us is the base is x and the exponent is m. We also refer to exponents as powers.
And notice there that we have x times x times x. Until we have as many factors as there is m. Let's look at some examples.
x squared or x to the second power. x cubed or x to the third power. And we have b to the fourth power.
Combining exponents. In order to add or subtract you must. have like terms. Which means you've got to add apples to apples and oranges to oranges.
X's go with X's. B's go with B's. 3's go with 3's. You can't mix them up. So if you look at this example here we have X to the M plus X to the N plus X to the M.
We can only add the M's together. And so we have 2 X to the M power plus X to the N power. Now for the laws of exponents.
The product of powers. When multiplying with exponents, add the exponents. So we see we have x squared times x cubed.
2 plus 3 is 5 and so the answer is x to the 5th power. We add the exponents. Let's check out the proof. x squared times x cubed is x to the fifth power. So we'll expand them.
And after expanding both of them, we'll bring them together. And note that it's x to the fifth power. Just add the exponents. Our next part here is the power of a product.
Apply the power to each factor and simplify. So we have quantity 3ab squared, which means then 3 squared, a squared, b squared, and is 9a squared, b squared, when we simplify it. Pretty straightforward.
Great tool to have and to remember when we're dealing with higher orders later in Algebra 2. Power of a power. When raising an exponent to a power multiply the exponents. X squared quantity x squared to the third power. X squared times 3 x to the sixth power. Let's look at the proof of that.
Okay write it out in expanded form. There we go. Quotient of powers. When dividing with exponents, subtract the exponents. x cubed divided by x squared is the same as x to the third minus two, which is x to the one, which of course is x because we don't write ones, at least not when we're multiplying.
Proof. We'll expand it out. Is that true? Well, let's look here. Assemble our fraction Love that.
Cross them out and look what's left. Just one x. So it's true. Subtract the exponents. Power of a quotient.
Find the power of the numerator and the denominator and simplify. 2a quantity. 2a over b cubed.
In this case now we need to remember that the denominator cannot be zero. And so 2 to the third power, a to the third power, and b to the third power the power applies to everything inside the parentheses and there we go simplified ready to go zero exponent this is very important rule anything raised to the zero power equals one anything raised to the zero power equals one keep that in mind very important skill and thing to remember later in higher mathematics so x to the zero you equals one x y quantity x y to zero equals one proof i love this proof one over one is one x over x is one and even a thousand over a thousand is one i think you get the idea so x cubed over x cubed is one And so x to the 3 minus 3 is x to the 0. Therefore, it must be 1. Pretty easy using the rules of substitution and reflexive. Anything to the 0 power equals 1. Negative exponents. Very important you pay attention to this rule. A negative power is a reciprocal.
a to the negative 2 is the same thing as 1 over a squared. Notice also the negative sign is not there anymore. It's the negative exponent that means that this is a fraction, a reciprocal.
And 1 over a to the negative 2, well, that's also a reciprocal. Therefore, it equals a squared. And let's remember, denominators cannot be zero. Can't divide by zero, people.
Let's look at the proof of that last one. So we look at it, there's a compound fraction, and a fraction is a division problem. So when you divide by a fraction, you flip the fraction and multiply.
And of course, that's the fraction in the denominator. Voila, proof. Recap, we defined exponents, we stated the laws of exponents, we learned how to use exponents.
And you should have it all together. If you have any questions, be sure to send an email to mrcozzy at mrcozzy.com. And I hope you have a great day.