Given the following expression:
[tex]11^m=13^n[/tex]Solving :
[tex]\sqrt[n]{11^m}=\sqrt[n]{13^n}[/tex][tex]\sqrt[n]{11^m}=13[/tex]We can write the root in the following way:
[tex]\sqrt[n]{11^m}=11^{\frac{m}{n}}[/tex]Therefore:
[tex]11^{\frac{m}{n}}=13[/tex]If m>n :
[tex]11^{\frac{m}{n}}>11[/tex]If m[tex]11^{\frac{m}{n}}<11[/tex]Therefore, m have to be greater than n.
Finally: if m and n are integers:
[tex]11^{\frac{m}{n}}=11^{\frac{3}{2}}=36.48[/tex]To get a value equal to 13, n has to be close to m, otherwise the number will be very large.
In this case: (Using the same numbers of the example):
[tex]11^{\frac{m}{n}}=11^{\frac{3}{2.8}}=13.0550[/tex]Answer: It is impossible to find non-zero integer numbers, because m or n have to be a rational number.
