Question

simplifique

$log8 \sqrt{64 {}^{5} } + log3 \sqrt[5]{27 {}^{2} }$
​

Answer 1

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Logaritmo da potência

$\huge\boxed{\boxed{\boxed{\boxed{\sf\ell og_ba^n=n\cdot\ell og_ba}}}}$

Logaritmo do produto

$\huge\boxed{\boxed{\boxed{\boxed{\sf\ell og_c(a\cdot b)=\ell og_ca+\ell og_cb}}}}$

Potência de expoente fracionário

$\huge\boxed{\boxed{\boxed{\boxed{\sqrt[\sf n]{\sf a^m}=\sf a^{\sf\frac{m}{n}}}}}}$

$\underline{\tt c\acute alculos~auxiliares:}\\\sf \sqrt{64^5}=\sqrt{(2^6)^5}=\sqrt{2^{30}}\\\sf \sqrt{64^5}=2^{\frac{30}{2}}=2^{15}\\\sf\sqrt[\sf5]{\sf27^2}=\sqrt[\sf5]{\sf(3^3)^2}=\sqrt[\sf5]{\sf 3^6}=3^{\frac{6}{5}}$

$\sf \ell og_8\sqrt{64^5}+\ell og_3\sqrt[\sf 5]{\sf 27^2}=\ell og_{2^3}2^{15}+\ell og_33^{\frac{6}{5}}\\\sf\ell og_8\sqrt{64^5}+\ell og_3\sqrt[\sf 5]{\sf 27^2}=15\cdot\dfrac{1}{3}\ell og_22+\dfrac{6}{5}\ell og_33\\\sf\ell og_8\sqrt{64^5}+\ell og_3\sqrt[\sf 5]{\sf27^2}=5\cdot1+\dfrac{6}{5}\cdot 1=5+\dfrac{6}{5}\\\large\boxed{\boxed{\boxed{\boxed{\sf\ell og_8\sqrt{64^5}+\ell og_3\sqrt[\sf 5]{\sf27^2}=\dfrac{31}{25}}}}}\checkmark$