[MUSIC PLAYING] Let's look at our objectives for today. We'll start by defining a monomial. We'll then look at multiplying monomials with and without coefficients. We'll define rational exponents. And finally, we'll look at the product property with rational exponents.
Let's start by defining a monomial. Monomials are exponential expressions with non-negative integer exponents. Here is an example of a monomial, 5 times x to the third. There are three parts to a monomial-- the base, the coefficient, and the exponent. The base is repeatedly multiplied by itself according to the exponent.
Here, the base is x. The exponent indicates how many times the base is used in repeated multiplication. So here, the exponent is 3 and the coefficient is the number that acts as a multiplier to the base and the exponent. So here, the coefficient is 5. It's important to mention that the exponent is only being applied to the base and not the coefficient.
Now, let's look at how to multiply monomials with and without coefficients. We'll start with an example without coefficients. We have x squared times x to the fourth.
To simplify, we expand the expression using repeated multiplication. So x squared becomes x times x, and x to the fourth becomes x times x times x times x. Multiplying the x terms together gives us x to the sixth.
We can also determine the exponent of the product, x squared times x to the fourth, by adding the two exponents, 2 plus 4 together, to give us x to the sixth. So this is an example of the product property for exponential expressions, which works for the product of all monomials when the bases of the monomial are the same. So in general, we have x to the a times x to the b is equal to x to the a plus b.
Now, let's look at how we multiply monomials with coefficients. Here is an example. We have negative 3x times 5x to the sixth. We can use the commutative property of multiplication, which allows us to group our coefficients and variables together.
Multiplication is commutative because we can multiply numbers in any order. So grouping our coefficients and our x terms together gives us negative 3 times 5 times x times x to the sixth. Negative 3 times 5 gives us negative 15, and x times x to the sixth is the same as x to the first times x to the sixth. Adding our exponents using the product property for exponents gives us x to the seventh, so our final answer is negative 15 times x to the seventh.
Now, let's introduce rational exponents. Exponents are generally integers or fractions, but they may be any number, such as a decimal. A rational exponent is an exponent that can be represented as a fraction.
The product property applies to all types of exponents, including integers and infractions. Therefore, we add fractions when applying to rational exponents. We can easily add fractions when we have common denominators.
We first write equivalent fractions with a common denominator. We then add the numerators-- leaving the denominator unchanged. And finally, we reduce the fraction, if possible, by canceling out common factors in the numerator and denominator.
Now, let's do some examples using the product property with rational exponents. We want to simplify 2x to the 1/2 times 4x to the 5/2. We start by multiplying our coefficients, 2 times 4. And then grouping our x terms together, we multiply x to the 1/2 times x to the 5/2.
We need to add our exponential, meaning we need to add the fractions together. Our denominator's already the same, 2. So we can simply add our numerators together. 1 plus 5 is 6, and our denominator stays the same. 6 and 2 have a common factor of 2, so we can cancel it out by dividing the numerator and the denominator by 2, which gives us 3 over 1, which is 3. So our expression becomes 8 times x to the third.
Here's our last example. We want to simplify negative x to the 3/4 times 7x to the 110. We start by multiplying our coefficient. The negative in front of the x is the same as a negative coefficient. So we have negative 1 times 7, which is negative 7. We then group our x terms together and multiply x to the 3/4 times x to the 110.
Here, when we add our fractions together, we notice that our denominator is not the same. So we need to think of the least common denominator. The least common denominator of 4 and 10 is 20. To get a denominator of 20 in the first fraction, we multiply by 5 in the denominator and the numerator.
To get a denominator of 20 in the second fraction, we multiply by 2 in the denominator and the numerator. This gives us 15 over 20 plus 2 over 20. We now have a common denominator, so we add our numerators, which is 17, and our denominator is 20. So our fraction is 17 over 20, which is fully simplified because 17 and 20 have no other common factor than 1. So our final expression is negative 7 times x to the 17/20.
Let's go over our important points from today. Make sure you get these in your notes so you can refer to them later. Monomials are exponential expressions with non-negative integer exponents. There are three parts to a monomial-- the base, the coefficient, and the exponent. The product property of exponential expression says that the product of two monomials with the same base can be simplified by adding the original exponents.
So I hope that these key points and examples helped you understand a little bit more about the product property of exponents. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.
An exponential expression with non-negative integer exponents.