Hi!
Square roots where the things under the root are multiplied (for example [tex] \sqrt{xy} [/tex]) can be split up into [tex] \sqrt{x} * \sqrt{y} [/tex].
That's what I'm going to do in this case. I'm going to split [tex] \sqrt[3]{125x^{12}} [/tex] into [tex] \sqrt[3]{125} * \sqrt[3]{x^{12}} [/tex]
Now, the cube root of 125 is 5, because 5 * 5 * 5 = 125, and that is the definition of a cube root. (What multiplied 3 times by itself is equal to what's under the root)
Now to look at what the cube root of [tex] x^{12} [/tex] is.
A cube root can be rewritten as [tex] x^{\frac{1}{3}} [/tex], same as a square root can be rewritten to [tex] x^{\frac{1}{2}} [/tex].
So if you take the cube root of x^12, you can change it to:
[tex] (x^{12} )^{1/3} [/tex]
If you have a case like this, then you can multiply the two powers (in this case 12 and 1/3) to simplify it. 12 * 1/3 = 4, so you get x^4.
Now putting the two together, you get:
5[tex] x^{4} [/tex] as your answer.
Hope this helped!
