Question Video: Manipulating Exponential Expressions Mathematics

Video Transcript

If the expression two over three to the power of negative two 𝑦 is rewritten in the form 𝑃 to the power of 𝑦, what is the value of 𝑃?

So first of all, let’s consider what two over three or two-thirds to the power of negative two 𝑦 means. Well, to do this, what we’re gonna use is some exponent rules. Well, the first rule we’re gonna use is the fact that if we have 𝑥 to the power of 𝑎 and then this is to the power of 𝑏, then this is equal to 𝑥 to the power of 𝑎 multiplied by 𝑏. And in fact, we’re gonna use this in reverse because what we can do is rewrite our expression two over three or two-thirds to the power of negative two 𝑦 as two over three to the power of negative two. And then this is all to the power of 𝑦.

And the reason we’ve done this is because what we’re looking for is to get our expression in the form 𝑃 to the power of 𝑦, so something to the power of 𝑦. Well, next, what we’re gonna do is use another one of our exponent rules. And that is, if we have 𝑥 over 𝑦 to the power of negative 𝑎, then this is equal to the reciprocal of our fraction, so 𝑦 over 𝑥 to the power of 𝑎. And what this is is an adaptation of the exponent rule, which tells us that 𝑥 to the power of negative 𝑎 is equal to one over 𝑥 to the power of 𝑎.

Okay, great. So let’s apply this to our problem we have. So when we do that, what we have now is the reciprocal of two-thirds or two over three, which is three over two or three-halves, all to power of two. And then this is to the power of 𝑦. Well, if we get back to the question, what we were looking for is we want the expression rewritten in the form 𝑃 to the power of 𝑦. Well, we have that because we have something to the power of 𝑦. So now, what we need to do is identify what the value of 𝑃 is. So therefore, we can say that 𝑃 is gonna be equal to three over two or three-halves all squared. It’s also worth noting that we could’ve written this as nine over four.