Simplifying radicals (higher-index roots)
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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.