"Is it possible to find nonzero whole numbers m and n such that 11^m = 13^n? Explain."

Question

asked Feb 6, 2023 6.5k views

1 Answer

Wesley Egbertsen · Answer 1 · 2023-02-10T17:04:31+0000

Given the following expression:

Solving :

$\sqrt[n]{11^m}=\sqrt[n]{13^n}$
$\sqrt[n]{11^m}=13$

We can write the root in the following way:

$\sqrt[n]{11^m}=11^{(m)/(n)}$

Therefore:

$11^{(m)/(n)}=13$

If m>n :

$11^{(m)/(n)}>11$

If m
$11^{(m)/(n)}<11$ Therefore, m have to be greater than n.

Finally: if m and n are integers:

$11^{(m)/(n)}=11^{(3)/(2)}=36.48$

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):

$11^{(m)/(n)}=11^{(3)/(2.8)}=13.0550$

Answer: It is impossible to find non-zero integer numbers, because m or n have to be a rational number.