Rewriting expressions with exponents challenge 1 | Algebra 1 (TX TEKS) | Khan Academy

So we have this pretty complicated, some would say hairy, expression right over here. What I want you to do is pause this video and see if you can simplify this based on what you know about exponent rules.

All right, now let's do this together. There's many ways you could approach this, but what my brain wants to do is first try to simplify this part right over here. I have a bunch of stuff in here to an exponent power, and one way to think about that is if I have, let's say, A * B to the let's call it C power, this is the same thing as A to the C times B to the C power. So we could do that with this part right over here.

Actually, let me just simplify this so I don't have to keep rewriting things. So this can be rewritten as five M—or let me be careful—this is going to be 5^2 A R times M to the -13 2 A R times N^2, which is the same thing as 25.

Now, if I raise something to an exponent and then raise that to an exponent, there’s another exponent property here. If I have A to the B and then I raise that to the C, then I multiply the exponents; this is equal to A to the B times C power. So here we would multiply these exponents: 25 M^2 * -1/3 is -23, and then, of course, we have this N^2 right over here.

So actually, let me just rewrite everything so we don't lose too much track. So we have 75—I wrote M—75 M to the 1/3 N to the -7, and then I simplified the bottom part. I'll do that same color as 25 M to the -23 N^2.

Now, some of y'all might immediately be able to skip some steps here, but I'll try to make it very, very explicit. What I'm going to do is rewrite this expression as the product of fractions or as a product of rational expressions. So I could rewrite this as being equal to 75 / 25, which I think you know what that is, but I'll just write it like that, times—and then we’ll worry about these right over here—times M to the 1/3 over M to the -23, and then times—in blue—N to the -7 over N^2.

Now, 75 over 25 we know what that is; that’s going to be equal to 3. But how do we simplify this right over here? Well, here we can remind ourselves of another exponent property. If I have, let’s call it A, A to the B over C to the D actually has to have the same base over A to the C. This is going to be the same thing as A to the B minus C power.

So I can rewrite all of this business. I have my 3 here: 3 times M to the 1/3, and then I'm going to subtract this exponent. We have to be very careful; we're subtracting a negative, so we're subtracting -23. That's all that exponent for M, and then we're going to have times N to the -7 power minus 2.

And so now we are in the home stretch. This is going to be equal to 3 * M to the—what’s 1/3 - -2/3? Well, that’s the same thing as 1/3 + 2/3, which is just 3/3, which is just 1. So this is just M to the first power, which is the same thing as just M, and then that is going to be times -7 - 2; that is -9. So times N to the -9th power, and we are done.

That is strangely satisfying to take something that hairy and make it, I guess, less hairy. Now, some folks might not like having a negative 9 exponent here; they might want only positive exponents. So you could actually rewrite this, and we could debate whether it's actually simpler or less simple.

But we also know the exponent properties that if I have A to the -N, that is the same thing as 1 over A to the N. So based on that, I could also rewrite this as 3—we do the same color as that—3 as 3 times M, and then instead of saying times N to the 9, we could say that is over N to the 9th. So that's another way to rewrite that expression.