HELP! I tried multiple ways but still couldn't figure out how to solve this!!

$P = 5^{4/13}$

Therefore, a + b + c = 5 + 4 + 13 = 22.

Simplify the following:
5^(1/5) 25^(1/25) 125^(1/125) 625^(1/625)

625^(1/625) = (5^4)^(1/625):

5^(1/5) 25^(1/25) 125^(1/125)×5^(4/625)

125^(1/125) = (5^3)^(1/125):

5^(1/5) 25^(1/25)×5^(3/125)×5^(4/625)

25^(1/25) = (5^2)^(1/25):

5^(1/5)×5^(2/25)×5^(3/125)×5^(4/625)

5^(1/5)×5^(2/25)×5^(3/125)×5^(4/625) = 5^(1/5 + 2/25 + 3/125 + 4/625):

5^(1/5 + 2/25 + 3/125 + 4/625)

Put 1/5 + 2/25 + 3/125 + 4/625 over the common denominator 625. 1/5 + 2/25 + 3/125 + 4/625 = 125/625 + (25×2)/625 + (5×3)/625 + 4/625:

5^(125/625 + (25×2)/625 + (5×3)/625 + 4/625)

25×2 = 50:
5^(125/625 + 50/625 + (5×3)/625 + 4/625)

5×3 = 15:

5^(125/625 + 50/625 + 15/625 + 4/625)

125/625 + 50/625 + 15/625 + 4/625 = (125 + 50 + 15 + 4)/625:

5^((125 + 50 + 15 + 4)/625)

125 + 50 + 15 + 4 = 194:

5^(194/625)

Thanks guests :)

P = 5^{1/5} \cdot 25^{1/25} \cdot 125^{1/125} \cdot 625^{1/625} \dotsm

$P = 5^{1/5} \cdot 25^{1/25} \cdot 125^{1/125} \cdot 625^{1/625} \dotsm\\ P = 5^{1/5} \cdot 5^{2/5^2} \cdot 5^{3/5^3} \cdot 5^{4/5^4} \dotsm\\ P = 5^{(\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+\frac{4}{5^4}\dotsm\\)} \\ log_5 P = {(\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+\frac{4}{5^4}\dotsm)} \\ log_5 P = {(\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}\dotsm)} + {(\frac{1}{5^2}+\frac{1}{5^3}+\frac{1}{5^4}\dotsm)} +{(\frac{1}{5^3}+\frac{1}{5^4}+\frac{1}{5^5}\dotsm)+ \dots}\\ log_5 P =(\frac{1}{5}\div \frac{4}{5})+(\frac{1}{25}\div \frac{4}{5})+(\frac{1}{125}\div \frac{4}{5})+ \dots\\ log_5 P =(\frac{1}{5}\times \frac{5}{4})+(\frac{1}{25}\times\frac{5}{4})+(\frac{1}{125}\times\frac{5}{4})+ \dots\\ log_5 P =(\frac{1}{4})+(\frac{1}{20})+(\frac{1}{100})+ \dots\\ log_5 P =\frac{1}{4}\div \frac{4}{5}\\ log_5 P =\frac{1}{4}\div \frac{4}{5}\\ log_5 P =\frac{1}{4}\times \frac{5}{4}\\ log_5 P =\frac{5}{16}\\ P=5^{\frac{5}{16}}\\ P\approx 1.65359$

LaTex

P = 5^{1/5} \cdot 25^{1/25} \cdot 125^{1/125} \cdot 625^{1/625} \dotsm\\
P = 5^{1/5} \cdot 5^{2/5^2} \cdot 5^{3/5^3} \cdot 5^{4/5^4} \dotsm\\
P = 5^{(\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+\frac{4}{5^4}\dotsm\\)} \\
log_5 P = {(\frac{1}{5}+\frac{2}{5^2}+\frac{3}{5^3}+\frac{4}{5^4}\dotsm)} \\

log_5 P = {(\frac{1}{5}+\frac{1}{5^2}+\frac{1}{5^3}\dotsm)} 
+ {(\frac{1}{5^2}+\frac{1}{5^3}+\frac{1}{5^4}\dotsm)}
+{(\frac{1}{5^3}+\frac{1}{5^4}+\frac{1}{5^5}\dotsm)+ \dots}\\

log_5 P =(\frac{1}{5}\div \frac{4}{5})+(\frac{1}{25}\div \frac{4}{5})+(\frac{1}{125}\div \frac{4}{5})+ \dots\\
log_5 P =(\frac{1}{5}\times \frac{5}{4})+(\frac{1}{25}\times\frac{5}{4})+(\frac{1}{125}\times\frac{5}{4})+ \dots\\
log_5 P =(\frac{1}{4})+(\frac{1}{20})+(\frac{1}{100})+ \dots\\
log_5 P =\frac{1}{4}\div \frac{4}{5}\\
log_5 P =\frac{1}{4}\div \frac{4}{5}\\
log_5 P =\frac{1}{4}\times \frac{5}{4}\\
log_5 P =\frac{5}{16}\\
P=5^{\frac{5}{16}}\\
P\approx 1.65359

Hi Melody:

When multiplied together, you get the following number:

(5^(1/5)) * (25^(1/25)) * (125^(1/125)) * (625^(1/625)) ==1.64801169512666976301802035295......etc.

1 - Guest #1 answer ==5^(4/13) ==1.64084551246609057805181222444......etc.

2 - Guest #2 answer==5^(194/625)==1.64801169512666976301802035295....etc.

3 - Melody's answer ==5^(5/16) ==1.65359110076253520866303943290......etc.