[MUSIC PLAYING] Let's look at our objectives for today. We'll start by looking at the components of exponential expressions. We'll then introduce the product and the power properties of exponents. And finally, we'll do some examples multiplying terms using the product and power properties of exponents.
Now, let's review the components of exponential expressions. The term below, 9a to the third, has three components. The 9 is the coefficient, the a is the base, and the 3 is the exponent or power.
Now, let's review the product and power properties of exponents. The product property of exponents says x to the m times x to the n is equal to x to the m plus n. So for example, x squared times x to the seventh would be equal to x to the 2 plus 7, which equals x to the 9. So the product property of exponents says that, when we multiply exponential terms with the same base, we can add their exponents to simplify.
Our second property, the power property, of exponents says that x to the m to the n is equal to x to the m times n. For example, x to the fourth to the third is the same as x to the fourth multiplied by itself three times. We can then use the product property of exponents to add the fours together-- 4 plus 4 plus 4-- which is the same as 4 times 3, which is the same as 12. So x to the fourth to the third is equal to x to the 12th. Therefore, when taking the power of an exponential expression, the exponents are multiplied.
When using the power property, we sometimes need to multiply fractional exponents. This involves multiplying across the numerators and denominators of the fractions. Fractions are reduced by canceling common factors of the numerator and denominator.
Now, let's do some examples using the product and power properties of exponents. Here's our first example. We want to simplify x to the 1/2 to the 2/3. The power property of exponents can be used to multiply the two fractional exponents to get a single exponent. Therefore, we have x to the 1/2 times 2/3.
When we multiply our fractions, we multiply straight across-- numerator by the numerator and denominator by denominator. So we have x to the 2 over 6. 2/6 can be simplified because 2 is a common factor of both 2 and 6. So dividing the numerator and denominator by 2 gives us 1/3. So our final answer is x to the 1/3.
Here's our last example. We want to simplify the expression x to the 3/5 times x to the 7/10 all raised to the 1/2. We start by simplifying in our parentheses. We have two exponential terms multiplying together, and they both have a base of x. Therefore, we can use the product property of exponents and add our exponents together.
To add our fractions, we know we need a common denominator. The least common denominator of 5 and 10 is 10. So we multiply our first fraction by 2 in the denominator and the numerator, and we leave our second fraction unchanged. So we now have 6/10 plus 7/10.
Now that our denominator is the same, we can add our numerators. 6 and 7 gives us 13, and our denominator stays 10. So our expression is now x to the 13/10 to the 1/2. We then use the power property to multiply 13/10 and 1/2. We can multiply straight across-- numerator by numerator and denominator by denominator-- which gives us x to the 13/20. We don't need to simplify the fraction because 13 and 20 do not have any common factors other than 1, So our final answer is x to the 13/20.
Now, let's go over our important points from today. Make sure you get these in your notes so you can refer to them later. The three components of an exponential expression are coefficients, bases, and exponents. The product property of exponents says that, when multiplying exponential terms together with the same base, the exponents are added. And the power property of exponents says that, when taking the power of an exponential expression, the exponents are multiplied.
So I hope that these important points and examples helped you understand a little bit more about multiplying terms with exponent properties. Keep using your notes, and keep on practicing. And soon, you'll be a pro. Thanks for watching.
