Rewriting expressions with exponents challenge 1
About this video
The video simplifies 75 m^(1/3) n^(-7) divided by (5 m^(-1/3) n)^2 using exponent rules. It applies the power of a product and power of a power rules to get 25 m^(-2/3) n^2 in the denominator, then uses the quotient rule to reach 3 m n^(-9), which can also be written 3m / n^9.
You'll be able to
- use (ab)^c = a^c b^c and (a^b)^c = a^(bc) to simplify a power of a product.
- use a^b / a^c = a^(b - c), taking care when subtracting a negative exponent.
- rewrite a negative exponent as a fraction: a^(-n) = 1 / a^n.
Key words
- exponent
- The small raised number showing how many times a base is used as a factor.
- power of a power rule
- (a^b)^c = a^(bc): multiply the exponents.
- quotient rule
- a^b / a^c = a^(b - c) for the same base: subtract the exponents.
- negative exponent
- a^(-n) means 1 / a^n.
Transcript
- [Presenter] So we have this pretty complicated, some would say hairy, expression right over here. And 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, if I have, let's say, A times 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. And actually, let me just simplify this so I don't have to keep rewriting things. So this can be rewritten as 5M, or let me be careful. This is gonna be five squared times M to the negative one third squared times N squared, which is the same thing as 25.
Now if I raise something to an exponent and then raise that to an exponent, so there's another exponent property here. If I have A to the B, and then I raise that to the C, then I multiplied the exponents. This is equal to A to the B times C power. So here we would multiply these exponents.
So it's 25M. Two times negative one third is negative two thirds. And then of course we have this N squared right over here. So actually lemme just rewrite everything so we don't lose too much track. So we have 75. I wrote M. 75. M to the one third. N to the negative seven.
And then I simplified the bottom part. I'll do that same color. As 25. M to the negative two thirds N squared. 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 read, what I'm gonna do is rewrite this expression as the product of fractions or as a product of rational expression.
So I could rewrite this. This is being equal to 75 divided by 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 one third over M to the negative two thirds, and then times, I'll put this in blue.
Times N to the negative seven over N squared. Now 75 over 25, we know what that is, that's going to be equal to three. But how do we simplify this right over here? Well, here we can remind ourselves another exponent property. If I have, let's call it A to the B over C to the D, actually it 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 three here. Three times M to the one third. And then I'm gonna subtract this exponent. We have to be very careful. We're subtracting a negative.
So we're subtracting negative two thirds. That's all that exponent for M. And then we're going to have times N to the negative seven power minus two. And so now we are in the home stretch. This is going to be equal to three times M to the, what's one third minus negative two thirds?
Well, that's the same thing as one third plus two thirds, which is just three thirds, which is just one. 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 negative seven minus two, that is negative nine.
So times N to the negative ninth power. And we are done. And 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 nine 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 negative N, that is the same thing as one over A to the N. So based on that, I could also rewrite this as three.
We do the same color as that three. As three times M. And then instead of saying times N to the negative nine, we could say that is over, that is over, N to the ninth. So that's another way to rewrite that expression.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.