Video: Evaluating Algebraic Expressions Using Laws of Exponents

If 9^(π‘₯) = 4 and 9/9^(𝑦) = 5, what is the value of 9^(π‘₯ + 𝑦)?

02:40

Video Transcript

If nine to the power of π‘₯ equals four and nine over nine to the power of 𝑦 equals five, what is the value of nine to the power π‘₯ plus 𝑦?

In this question, we have two separate equations. And we’re asked to find the value of one term, nine to the power of π‘₯ plus 𝑦. We can see in these equations that we’re given that we do have something similar to nine to the power of π‘₯ plus 𝑦. In the first equation, we’re told nine to the power of π‘₯. And in the second equation, there’s a nine to the power of 𝑦. So let’s see if we can rearrange them and join the equations together.

To do this, we’re going to need to remember our rules of exponents. In particular, the rule π‘₯ to the power of π‘Ž times π‘₯ to the power of 𝑏 equals π‘₯ to the power of π‘Ž plus 𝑏. If we take a closer look at our term, nine to the power of π‘₯ plus 𝑦, we can see that it’s very similar to the term we’ve got in the rules of exponents, the π‘₯ to the power of π‘Ž plus 𝑏.

Since π‘₯ to the power of π‘Ž plus 𝑏 is equal to two terms, π‘₯ to the power of π‘Ž times π‘₯ to the power of 𝑏, then we can say that nine to the power of π‘₯ plus 𝑦 is also equal to nine to the power of π‘₯ times nine to the power of 𝑦. If we look at the equations we’re given, we can see that we do have an equation that’s equal to nine to the power of π‘₯. But we don’t yet have an equation that’s equal to nine to the power of 𝑦.

So let’s take our second equation and see if we can rearrange it so that nine to the power of 𝑦 is the subject of our equation. To start by rearranging our equation nine over nine to the power of 𝑦 equals five, we want to move nine to the power of 𝑦 from being on the denominator. And we can do this by multiplying both sides of our equation by nine to the power of 𝑦. This will give us nine equals five times nine to the power of 𝑦. And next, to find nine to the power of 𝑦 by itself, we need to divide both sides of our equation by five, giving us nine over five equals nine to the power of 𝑦.

So now we have the equations nine to the power of π‘₯ equals four and nine to the power of 𝑦 equals nine over five. We can substitute these into our equation nine to the power of π‘₯ plus 𝑦 equals nine to the power of π‘₯ times nine to the power of 𝑦. This will give us nine to the power of π‘₯ plus 𝑦 equals four times nine over five. And we can simplify the numerical values on the right-hand side to give us nine to the power of π‘₯ plus 𝑦 equals 36 over five.

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