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A worked example of calculating an expression that has both a radical and an exponent. In this example, we evaluate 6^(1/2)⋅(⁵√6)³. Created by Sal Khan.
Video transcript
Let's see if we can simplify 6 to the 1/2 power times the fifth root of 6 and all of that to the third power. And I encourage you to pause this video and try it on your own. So let me actually color code these exponents, just so we can keep track of them a little better. So that's the 1/2 power in blue. This is the fifth root here in magenta. And let's see. In green, let's think about this third power. So one way to think about this fifth root is that this is the exact same thing as raising this 6 to the 1/5 power, so let's write it like that. So this part right over here, we could rewrite as 6 to the 1/5 power, and then that whole thing gets raised to the third power. And of course, we have this 6 to the 1/2 power out here, 6 to the 1/2 power times all of this business right over here. Now, what happens if we raise something to an exponent and then raise that whole thing to another exponent? Well we've already seen in our exponent properties, that's the equivalent of raising this to the product of these two exponents. So this part right over here could be rewritten as 6 to the-- 3 times 1/5 is 3/5-- 6 to the 3/5 power. And of course, we're multiplying that times 6 to the 1/2 power. 6 to the 1/2 power times 6 to the 3/5 power. And now, if you're multiplying some base to this exponent and then the same base again to another exponent, we know that this is going to be the same thing. And actually we could put these equal signs the whole way, because these all equal each other. This is the same thing as 6 being raised to the 1/2 plus 3/5 power, 1/2 plus 3 over 5. Now, what's 1/2 plus 3 over 5? Well, we could find a common denominator. It would be 10, so that's the same thing as-- actually let me just write it this way-- this is the same thing as 6 to the-- instead of 1/2, we can write it as 5/10. Plus 3/5 is the same thing as 6/10 power, which is the same thing-- and we deserve a little bit of a drum roll here, this wasn't that long of a problem-- 6 to the 11/10 power. I'll just write it all, 11/10 power. And so, that looks pretty simplified to me. I guess we're done.