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# Evaluating mixed radicals and exponents

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.