How to simplify fraction exponents? For example $(x^3+4x^2+7)^{\frac{1}{4}}$

Question

I know that if the exponent $n$ is an integer, I can use Newton's Binom to simplify.

But I have no idea what to do if the exponent is a fraction.

My question is, how would you simplify $$(x^3+4x^2+7)^{\frac{1}{4}}$$, or $\sqrt[4]{(x^3+4x^2+7)}$ ? Are there any formulas for this?

Answer 1

You can rewrite a power of $1/4$ as a fourth root — that's absolutely correct. But there's no way to simplify further roots of sums or differences, unless you can factor it (in which case you'll be simplifying a root of a product, which is a whole different story).

Answer 2

one can write $$e^{\frac{\ln(x^3+4x^2+7)}{4}}$$