Zadanie 38

1) Uprośćmy $a$:
$a=\log_{\sqrt{5}}{2}\cdot\log_{2}{25}=\frac{\ln 2}{\ln \sqrt{5}}\cdot\frac{\ln 25}{\ln 2} =\frac{\ln 2}{\tfrac{1}{2}\ln 5}\cdot\frac{2\ln 5}{\ln 2}=4.$

Uprośćmy $b$:
$b=\dfrac{\log_{5}{6}}{\log_{5}{8}}=\dfrac{\ln 6}{\ln 8}=\log_{8}{6}.$ Zatem $b+1=\log_{8}{6}+1=\dfrac{\ln 6+\ln 8}{\ln 8}=\dfrac{\ln(48)}{\ln 8}=\log_{8}{48}.$

Obliczamy $a^{b+1}$, korzystając z $a^{\log_{b}c}=c^{\log_{b}a}$:
$a^{b+1}=4^{\log_{8}48}=48^{\log_{8}4}.$
$\log_{8}4=\dfrac{\ln 4}{\ln 8}=\dfrac{2\ln 2}{3\ln 2}=\dfrac{2}{3}$, więc
$48^{\log_{8}4}=48^{2/3}=(48)^{2/3}=(8\cdot 6)^{2/3}=8^{2/3}\cdot 6^{2/3}=4\cdot 6^{2/3}=4\sqrt[3]{36}.$

Odpowiedź: $a^{b+1}=4\cdot 6^{2/3}=4\sqrt[3]{36}$.