Simplify.(3x4y2)3\left(\frac{3x^4}{y^2}\right)^3

27x12y6\frac{$$27x^{12}$$}{$$y^6$$}y627x12

Simplify. ( 3 x 4 y 2 ) 3 \left(\frac{3x^4}{y^2}\right)^3

Everybody. So let's check out this next practice problem here. I know it looks kinda scary at first. It looks really daunting. We have lots of numbers, exponents, and multiplication and division happening, but the most important thing you can try to do here is just go slow, go one step at a time. Lots of students will make mistakes because they'll try to do way too many steps altogether in one step. The best thing you can do is just try to break it up and deal with one thing at a time. I'm gonna help help you go ahead and break down this problem. Let's get started here. We have a quotient, three x to the fourth in the numerator over y squared in the denominator, and that entire thing is raised to an exponent. So typically with these types of problems, it's best to work from the outside in. So what I mean by this is the very first thing that we see here is we have a quotient that is raised to a power. So that tells us we're gonna use the power of a quotient rule first. We're gonna take this three on the outside. We distribute it to everything that's in the numerator, everything, the whole three x to the fourth power, and also everything in the denominator as well to the y squared. And then we're gonna go ahead and start simplifying one by one. Now in the numerator, because we have multiple terms like a three and an x to the fourth, we're gonna keep a parenthesis. We're gonna have three x to the fourth, and this is all raised to the third power. On the bottom, things are a little bit simpler because we only have one term. It's just a y squared. But because this has an exponent already, we're still just going to keep a parenthesis here. This is also going to be raised to the third power. Alright. So basically, we just distributed the quote, the exponent to the quotient. But now we can see here that we start simplifying. We have a power of a product rule. We have a three that is on the outside of a three X to the fourth power, so we distribute that exponent into the numerator, which is a product. So a lot again, a lot of times you're gonna use multiple rules inside of these problems. Again, I'm not worried about the denominator first, I'm just only working with the numerator. So if I distributed this three to the three coefficient, what I end up with is three raised to the third power, and then I can include parentheses here. This is really just x to the fourth power raised to the third power. Alright? So again, all I did was I just used the same sort of setup. I have a variable raised to an exponent with another exponent on the outside, so it's always good always a good idea to keep those parentheses there. Alright? That's the numerator. On the denominator, however, it's a little bit simpler because we didn't have a product here. What I do notice is that I have y squared raised to another power, which is the third power. This is a power of a power. So when we have variables raised to a power and we have another exponent on the outside, we multiply those exponents. Remember, this is like we have y squared times itself three times. So what you can do is you can multiply those exponents, y to the two times three power. Okay? Now one last thing here is, again, just work one at a time. We're gonna go back to the numerator here. Start with this one first. What is three to the third power? Can you simplify that further? Three to the third power is just a number. Right? It's just three multiplied by itself three times. Three times three times three. So So you're gonna work it out on your own off to the side. Three times three is nine, and nine times three is 27. So this three to the third power really just becomes 27 here in the bottom. Alright? Now what about the what about the x to the fourth raised to the third power? We're gonna use the exact same rule that we just used in the denominator. We have a power raised to another power, and so it's basically like we can just multiply those exponents four times three. Alright? So that's the numerator. On the denominator, we have y to the two times three power, which becomes y to the sixth. Alright? Now there's one final step we can do here, which is we just have to simplify these two exponents that get multiplied. So this is just 27 times x to the twelfth power divided by y to the sixth power. Alright. Is there anything else that we can do? Can we use any quotient rules, any product rules? Is there any sort of coefficients or numbers that we can sort of cancel out? There's nothing because 27, we don't have anything to cancel that out with. We have x to the twelfth, that's one variable. Y to the sixth, that's another variable, so you can't divide those exponents. And that's really all there is to it. Right? You can't really simplify this any further. So again, lots of problems will involve multiple rules and they may look really, really daunting at first. The most important thing you can do is just try to break it down and use one rule at a time. Don't try to do too much at the same time. Alright? So that's really all there is to it. I'll see you in the next one and thanks for watching.