Show how to simplify the following number by hand:

4^{log5/44}
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5^{log5/45}

First, rewrite the log_5/4 (4) in the numerator as log_4(4) / log_4(5/4). We can do the same in the denominator: log_5/4(5) = log_5(5)/log_5(5/4). This can now be rewritten as: 4 ^(1/log_4(5/4)) / 5^(1/log_5(5/4)). Both the logs contain a factor equal to their base, we can rewrite to:

4^(log_4(5)-1) / 5^(1/(1-log_5(4))).

Let's define x = log_4(5). Then we have that log_5(4) = 1/x, so we can rewrite the above as:

4^(1/(x-1)) / 5^(1/(1-(1/x))). Rewriting the latter term:

4^(1/(x-1)) / 5^(x/(x-1)). Now, from the definition of x, we know that 5 = 4^x, so we can rewrite everything in powers of 4:

4^(1/(x-1)) / 4^(x^2/(x-1)), or: 4^((1-x^2)/(x-1)).

As 1-x^2 = (1-x)*(1+x); we get: 4^(-(1+x)), or (4^(-1))*(4^(-x)). This is equal to 1/4 * 1/5 = 1/20.