Solving equations with rational exponents

0:00- [Instructor] So, let's start with 2 times the square root of y to the 3rd minus 53 is equal to 1,947. Pause this video. Have a go at that. All right, now let's work through this together. So, generally speaking, it would be good to isolate this business right over here on one side of the equation.

0:23So, the first thing that I want to do is I wanna get rid of this subtracting the 53, and I can do that by, on the left hand side, and I can do that by adding 53 to both sides. And so those cancel out and I am left with, on the left hand side, just 2 times the square root of y to the 3rd, and then that is going to be equal to 1,947 plus 53 is 2,000.

0:48So the numbers are working out. So, now, I can divide by 2 if I really want to isolate the radical of something to an exponent. So, let me just divide both sides by 2. And I get y the square root of y to the 3rd power is equal to 1,000. Now, how do I go about solving this?

1:06So, the way my brain likes to approach it is to rewrite this as a rational exponent. It's in the title of the video, or at least I think it will be the title of this video. So, this is the same thing. So, let me write this here. If I were to take the square root of something, that's the same thing as that something to the 1/2 power.

1:24So this is the same thing as y to the 3rd, to the 1/2 power is equal to 1,000. Or another way to think about it. We also know that if we raise something to a power and then raise that something to a power, you can take the product of these two exponents. So, this is the same thing as y to the 3/2 is equal to 1,000.

1:48So, we're making progress here using our exponent properties. But how do we solve this? And so here the key is, is we can raise both sides to the reciprocal of this exponent or the reciprocal power here. So, if we were to raise y to the 3/2 to the 2/3, that's the same thing as raising y to the first power 'cause 3/2 times 2/3 is 1.

2:13But we can't just do it to the left hand side. We have to do it to the right hand side as well. So, this is just the same thing as y to the 1st power. So, we get y is equal to, and what's 1,000 to the 2/3 power? Well, there's a lot of different ways you can approach this.

2:28The way I like to break it out is I could say that that is the same thing as 1,000 to the 1/3 and then I am going to square that. So, 1,000 to the 1/3 or what's the cube root of 1,000? What times itself times itself is 1,000? Well, this is 10. 10 times 10 times 10 is 1,000.

2:48So 10 squared is equal to 100. So y is equal to 100. Let us do another example here. Let's say that we have 3x to the 2.5th power, minus 16 is equal to 80. Pause the video and have a go at this. All right, so same idea. We wanna isolate the x to the 2/5. And actually, from the get-go I could rewrite 2/5 as 5/2.

3:18That makes it very clear. It's a rational exponent there. But let me add 16 to both sides. Get rid of that 16. I'm just trying to isolate the x term on one side. So if I add 16 to both sides, those cancel out and I am left with 3x to the 5/2 on the left hand side and on the right hand side I have 96.

3:43Now, I can divide both sides by 3 to isolate the X term. So then I have 96 divided by 3. So I have x to the 5/2 is equal to 32. And now, I'll do that same exact idea. I'll raise both sides to the reciprocal power. So, here I'm raising it to the 2/5 and I have to do that to both sides.

4:09I'm going to raise to the 2/5. So I get x is equal to. Now, I can rewrite this, at least this is how I like to do it, as 32 to the 1/5 and then that squared. What times itself times itself, times itself, times itself? I think I got that right. It's going to be 32.

4:26Or what to the 5th power is 32? Well, that's 2. 2 to the 5th power is 32. And then we have 2 squared is equal to 4. And we are done.