Question

reduza os radicais abaixo ao mesmo índice:​

Answer 1

Um radical pode ser escrito como um expoente fracionário como é mostrado abaixo:

$\rightarrow~~\sqrt[b]{a^c}~=~a^{\frac{c}{b}}$

Vamos então utilizar esta propriedade nas questões.

b)

$\sqrt[3]{7^2}~~e~~\sqrt[5]{7^4}~~=~~7^{\frac{3}{2}}~~e~~7^{\frac{4}{5}}\\\\\\MMC~dos~expoentes~=~10\\\\\\\sqrt[3]{7^2}~~e~~\sqrt[5]{7^4}~~=~~7^{\frac{5~.~3}{10}}~~e~~7^{\frac{2~.~4}{10}}\\\\\\\sqrt[3]{7^2}~~e~~\sqrt[5]{7^4}~~=~~7^{\frac{15}{10}}~~e~~7^{\frac{8}{10}}\\\\\\\boxed{\sqrt[3]{7^2}~~e~~\sqrt[5]{7^4}~~=~~\sqrt[10]{7^{15}}~~e~~\sqrt[10]{7^8}}$

c)

$\sqrt[8]{a^3}~~e~~\sqrt[]{a^5}~~=~~a^{\frac{3}{8}}~~e~~a^{\frac{5}{2}}\\\\\\MMC~dos~expoentes~=~8\\\\\\\sqrt[8]{a^3}~~e~~\sqrt[]{a^5}~~=~~a^{\frac{1~.~3}{8}}~~e~~a^{\frac{4~.~5}{8}}\\\\\\\sqrt[8]{a^3}~~e~~\sqrt[]{a^5}~~=~~a^{\frac{3}{8}}~~e~~a^{\frac{20}{8}}\\\\\\\boxed{\sqrt[8]{a^3}~~e~~\sqrt[]{a^5}~~=~~\sqrt[8]{a^{3}}~~e~~\sqrt[8]{a^{20}}}$

d)

$\sqrt[3]{2^2}~~e~~\sqrt[4]{2}~~=~~2^{\frac{2}{3}}~~e~~2^{\frac{1}{4}}\\\\\\MMC~dos~expoentes~=~12\\\\\\\sqrt[3]{2^2}~~e~~\sqrt[4]{2}~~=~~2^{\frac{4~.~2}{12}}~~e~~2^{\frac{3~.~1}{12}}\\\\\\\sqrt[3]{2^2}~~e~~\sqrt[4]{2}~~=~~2^{\frac{8}{12}}~~e~~2^{\frac{3}{12}}\\\\\\\boxed{\sqrt[3]{2^2}~~e~~\sqrt[4]{2}~~=~~\sqrt[12]{2^{8}}~~e~~\sqrt[12]{2^{3}}}$

e)

$\sqrt[4]{x}~~e~~\sqrt[5]{x^3}~~=~~x^{\frac{1}{4}}~~e~~x^{\frac{3}{5}}\\\\\\MMC~dos~expoentes~=~20\\\\\\\sqrt[4]{x}~~e~~\sqrt[5]{x^3}~~=~~x^{\frac{5~.~1}{20}}~~e~~x^{\frac{4~.~3}{20}}\\\\\\\sqrt[4]{x}~~e~~\sqrt[5]{x^3}~~=~~x^{\frac{5}{20}}~~e~~x^{\frac{12}{20}}\\\\\\\boxed{\sqrt[4]{x}~~e~~\sqrt[5]{x^3}~~=~~\sqrt[20]{x^{5}}~~e~~\sqrt[20]{x^{12}}}$

f)

$\sqrt[12]{7^5}~~e~~\sqrt[18]{7^5}~~e~~\sqrt[6]{7^5}~~=~~7^{\frac{5}{12}}~~e~~7^{\frac{5}{18}}~~e~~7^{\frac{5}{6}}\\\\\\MMC~dos~expoentes~=~36\\\\\\\sqrt[12]{7^5}~~e~~\sqrt[18]{7^5}~~e~~\sqrt[6]{7^5}~~=~~7^{\frac{3~.~5}{36}}~~e~~7^{\frac{2~.~5}{36}}~~e~~7^{\frac{6~.~5}{36}}\\\\\\\sqrt[12]{7^5}~~e~~\sqrt[18]{7^5}~~e~~\sqrt[6]{7^5}~~=~~7^{\frac{15}{36}}~~e~~7^{\frac{10}{36}}~~e~~x^{\frac{30}{36}}$

$\boxed{\sqrt[12]{7^5}~~e~~\sqrt[18]{7^5}~~e~~\sqrt[6]{7^5}~~=~~\sqrt[36]{7^{15}}~~e~~\sqrt[36]{7^{10}}~~e~~\sqrt[36]{7^{30}}}$