Rewriting roots as rational exponents | Mathematics I | High School Math | Khan Academy

We're asked to determine whether each expression is equivalent to the seventh root of v to the third power. And like always, pause the video and see if you can figure out which of these are equivalent to the seventh root of v to the third power.

Well, a good way to figure out things that are equivalent is to just try to get them all in the same form. So the seventh root of v to the third power, v to the third power, the seventh root of something is the same thing as raising it to the one-seventh power. So this is equivalent to v to the third power raised to the one-seventh power.

If I raise something to an exponent and then raise that to an exponent, well, then that's the same thing as raising it to the product of these two exponents. So this is going to be the same thing as v to the 3 times 1/7 power, which of course is 3/7. So we've written it in multiple forms. Now let's see which of these match.

So v to the third to the one-seventh power, well that was the form that we have right over here, so that is equivalent to v to the three-sevenths. That's what we have right over here, so that one is definitely equivalent. Now let's think about this one: this is the cube root of v to the seventh. Is this going to be equivalent?

Well, one way to think about it is this is going to be the same thing as v to the one-third power. Actually, no, this wasn't the cube root of v to the seven; this was the cube root of v and that to the seventh power. So that's the same thing as v to the one-third power and then that to the seventh power.

So that is the same thing as v to the seven-thirds power, which is clearly different than v to the three-sevenths power. So this is not going to be equivalent for all v's for which this expression is defined.

Let's do a few more of these or similar types of problems dealing with roots and fractional exponents. The following equation is true for g greater than or equal to zero and d is a constant. What is the value of d?

Well, if I'm taking the sixth root of something, that's the same thing as raising it to the one-sixth power. So the sixth root of g to the fifth is the same thing as g to the fifth raised to the one-sixth power.

Just like we just saw in the last example, that's the same thing as g to the 5 times 1/6 power. This is just our exponent properties: if I raise something to an exponent and then raise that whole thing to another exponent, I can just multiply the exponents. So that's the same thing as g to the 5/6 power, and so d is 5/6. The sixth root of g to the fifth is the same thing as g to the five-sixths power.

Let's do one more of these. The following equation is true for x greater than zero and d is a constant. What is the value of d?

All right, this is interesting, and I forgot to tell you in the last one, but pause this video as well and see if you can work it out on—or pause for this question as well and see if you can work it out.

Well here, let's just start rewriting the root as an exponent. I can rewrite the whole thing; this is the same thing as 1 over, instead of writing the seventh root of x, I'll write x to the 1/7 power is equal to x to the d.

If I have 1 over something to a power, that's the same thing as that something raised to the negative of that power. So that is the same thing as x to the negative 1/7 power, and so that is going to be equal to x to the d.

So d must be equal to—d must be equal to negative 1/7. So the key here is when you're taking the reciprocal of something, that's the same thing as raising it to the negative of that exponent. Another way of thinking about it is you could view this as x to the one-seventh to the negative one power, and then if you multiply these exponents, you get what we have right over there. But either way, d is equal to negative 1/7.