What is the largest $n$ such that $a = 2^{306} \cdot 3^{340}$ is a perfect $n$th power?
Hint: it will be \(\sqrt{2^{306}} \times \sqrt {3^{340}}\), that way \(n^2 = {2^{306}} \times {3^{340}}\)
No we are asked to find the $n$ for which is the square of that. We need the largest *n* th root.
n= Greatest Common Divisor of 306 and 340
306=2×3^2×17
340=2^2×5×17
n=GCD(306,340)=2×17=34
a=2^306×3^340=2^(9×34)×3^(10×34)=(2^9)^34×(3^10)^34
a=(2^9×3^10)^34=30,233,088^34
n=34