Assume $a_i, b_i$ are positive reals for $i = 1,2,...,n.$ Show that $$(\prod_{i=1}^{n} (a_i+b_i))^{\frac{1}{n}} \geq (\prod_{i=1}^{n} {a_i})^{\frac{1}{n}}+(\prod_{i=1}^{n} {b_i})^{\frac{1}{n}}.$$
I tried applying AM-GM to this product in some way, seeing that there is a product of $n$ terms and an exponent of $\frac{1}{n}.$ However, applying it directly only gives $$\frac{1}{n}\sum_{i=1}^{n} (a_i+b_i) \geq (\prod_{i=1}^{n} (a_i+b_i))^{\frac{1}{n}}$$ and $$\frac{1}{n}\sum_{i=1}^{n} a_i + \frac{1}{n}\sum_{i=1}^{n} b_i\geq (\prod_{i=1}^{n} {a_i})^{\frac{1}{n}}+(\prod_{i=1}^{n} {b_i})^{\frac{1}{n}},$$ which isn't helpful whatsoever. I can't think of other helpful inequalities that deal with sequence products like this, so I'm currently stuck trying to manipulate the expression and apply AM-GM somehow, but can't find a good way.
