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CCSS Math: HSN.RN.A.2

Simplify the cube root
of 125 x to the sixth y to the third power. So taking the cube
root of something is the same thing as raising
that something to the 1/3 power. So this is equal to
125 x to the sixth y to the third power
raised to the 1/3 power. And if we take a product
of a bunch of stuff and raise that to
the 1/3 power, that's the same thing as individually
raising each of the things to the 1/3 power and
then taking the product. So this is going
to be equal to 125 to the 1/3 power times x to the
sixth to the 1/3 power times y to the third to the 1/3 power. And then we can think about how
we can simplify each of these. What's 125 to the 1/3? Well, let's just
factor and see if we can have at least three
prime factors of something and maybe more than
one prime factor that shows up three times. So 125 is 5 times 25. 25 is 5 times 5. So 125 really is
5 times 5 times 5. So if you multiply 5 times
itself three times you get 125. 125 to the 1/3 power
is going to be 5. So this is going to
simplify to 5 times. And then x to the sixth
to the 1/3 power-- we saw this in a
previous example-- if you raise a
base to an exponent and then raise that whole
thing to another exponent, you can take the product
of the two exponents. So 6 times 1/3 is 6/3 or 2. So this part right
over here simplifies to x to the sixth divided
by 3 power or x squared. And then finally over
here, same principle. Raising y to the third power,
and then that to the 1/3 power. So that's going to
be y to the 3 times 1/3 power, or y to
the first power. And then times y. And we are done. And if you don't want to write
this little multiplication here, you could just write
this as 5x squared y. And we have simplified.