Question

Simplify: ln [(3x^3 + sin2x)^(1/3)]

Answer 1

Much of it depends on what you regard as simple...
\[\Large \ln\left[(3x^3 + \sin \ 2x )^{\frac13}\right]\]

Answer 2

For one thing, you can bring the exponent down, as per the properties of logarithms...
\[\Large \frac13\ln(3x^3 + \sin \ 2x )\]

Answer 3

I don't see how it can be simplified further from here...

Answer 4

That's the answer that the teacher gave, but I don't understand why the 1/3 is put in front of the ln, rather than distributed within the argument.

Answer 5

You can do that, but isn't what I'd call "simplify", since the expression would look more complicated.

Also, one does not simply distribute a fractional exponent, it'd be like
\[\Large \sqrt[3]{3x^3+ \sin \ 2x}\] and we don't really distribute the radical...

Answer 6

I don't know why my brain won't accept that you can't simply distribute an exponent within a parenthetical expression. Thanks for the explanation.

Answer 7

No problem :)