Rewriting roots as rational exponents

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 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 and 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 three times 1/7 power which of course is three sevenths three sevenths 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 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 R is this going to be equivalent well one way to think about it 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 a 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 7/3 power which is clearly different the two V to the 3/7 power so this is not going to be equivalent for all these all these for which this expression is defined let's do a few more of these or similar types of problems dealing with with with roots and 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 six the root of G to the fifth it's the same thing as G to the fifth raised to the one sixth power and just like we just saw in the last example that's the same thing as G to the five times one sixth power this is just our exponent properties 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 five six power and so D is five six five over six the sixth root of G to the fifth is the same thing as G to the five sixth 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 but pause this video as well and see if you can work it out on pause for this question as well and see if you can work it out well here well let's just start rewriting the root as an exponent so let me I can rewrite the whole thing this is the same thing as 1 over instead of writing the 7th root of X I'll write X to the one seventh power is equal to X to the D and if I have 1 over something to a power that's the same thing is 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 and 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 you could view this as you could view it 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