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Rules of Exponents - Problem 4
Simplifying an ugly expression using the rules of exponents. So what we have here is our two fractions both to powers and what we're going to do is try to combine these down to make one more simple fraction. And what we have to do basically first thing is to distribute the powers into each fraction and doing this I'm going to skip a couple steps, I'll walk you through what I'm doing, but hopefully you can follow along.
So the first thing I know is that this power and this -3 has to get distributed to every single term in this fraction. So it has to go to the 2, the x, the y and the z as well and I'm actually going to do all this stuff in my head and I'll talk you through what's going on.
So what we end up with is -2 to the -3. The -3 power is going to flip it down to the bottom, so we end up with 2 to the third in the denominator, -2 to the third is just -8. So this -2 in the numerator is going to turn into a -8 in the denominator.
X² to the negative third. We have a power to a power which means we have to multiply which we end up with x to the negative 6th. X to the negative 6th would be here, the negative is then going to put it in the bottom so we end up with an x to the 6th in the bottom. Y to the negative third, just distributing that in negative again tells us to go to the bottom so we have a y to the third in the denominator.
Our last one is z to the negative third in the denominator, negative third to the negative third, we multiply and we end up with z to the 9th. That's actually going to stay in the denominator as well because it started in the denominator, it's going to end up being to a positive power, so it's going to stay where it is. Another way you can think of that is this -3 is going to move it up to the numerator inside of the fraction, then this -3 in the outside would move it directly back down. So what we ended up with here is z to the 9th, let me make that 9 a little bit bigger for you, in the denominator and there's actually no numerator for this particular fraction here.
Moving on to our fraction, this is a giant multiplication. Doing the same thing we have -3² and negative squared is going to be positive so this turns into 9, z² squared, a power to a power multiply z to the 4th. X to the 4th squared again power to power we multiply giving us x to the eighth and then y to the first squared power to power end up with y².
So what we've managed to do is take two expressions both to powers and expanded them out to get one large fraction. Now all we have to do is combine like terms. So let me see if I have room, yeah I should be able to squeeze it over here. Looking at our numeric values we have 9 and -8 nothing we can do there so we end up with 9, 8 and a negative sign.
Looking at our xs, x to the 6th, x to the 8th both in the denominator when we're multiplying we just add our exponents so this ends up being x to the 14th, y³, y² again multiplying we just add our exponents so this ends up being y to the 5th and then z to the 4th over z to the 9th when we are dividing, we subtract the z to the 9th in the denominator is bigger which tells me that the difference 4 minus 9, 5 is going to be in the bottom, ending up with z to the 5th.
So by using our rules of exponents, we were able to expand both of these fraction and make them so we have all positive exponents which are easier to deal with, combine like terms ending up with our end result.