Question

Detalhe os procedimentos utilizados para resolver as seguintes operações com radicais (cada uma).
Efetue e simplifique:
a) (5√10)²
b) (4√2m-1)²
c) 7√450 - 4√320 + 3√80 - 5√800 - 20√5
d) √5 (√3 - √5 + √8)

(melhor resposta para quem detalhar como faz cada operação certinho) :)

Answer 1

$\Large\boxed{\begin{array}{l}\underline{\rm Produto\,de\,bases\,diferentes}\\\underline{\rm tomados\,ao\,mesmo\,expoente}\\\sf (a\cdot b)^n=a^n\cdot b^n\\\underline{\rm Quadrado\,da\,soma\,de\,dois\,termos}\\\sf (a+b)^2=a^2+2ab+b^2\\\underline{\rm Propriedade\,do\,radical\,aritm\acute etico}\\\sf \sqrt[\sf n]{\sf a^n}=a,\forall\,a\geqslant0\\\underline{\rm Produto\,de\,radicais\,de\,mesmo\,\acute indice}\\\sf \sqrt[\sf n]{\sf a}\cdot\sqrt[\sf n]{\sf b}=\sqrt[\sf n]{\sf a\cdot b}\end{array}}$

$\large\boxed{\begin{array}{l}\rm a)~\sf(5\sqrt{10})^2=5^2\cdot(\sqrt{10})^2=5^2\cdot\sqrt{10^2}\\\sf=25\cdot10=250\\\rm b)~\sf(4\sqrt{2}m-1)^2=(4\sqrt{2}m)^2-2\cdot4\sqrt{2}m\cdot1+1^2\\\sf(4\sqrt{2}m-1)^2=4^2\cdot(\sqrt{2})^2\cdot m^2-8\sqrt{2}m+1\\\sf (4\sqrt{2}m-1)^2=16\cdot2\cdot m^2-8\sqrt{2}m+1\\\sf (4\sqrt{2}m-1)^2=32m^2-8\sqrt{2}m+1\end{array}}$

$\large\boxed{\begin{array}{l}\begin{array}{c|c}\sf450&\sf2\\\sf225&\sf3\\\sf75&\sf3\\\sf25&\sf5\\\sf5&\sf5\\\sf1\\\sf\end{array}\\\sf 450=2\cdot3^2\cdot5^2\\\begin{array}{c|c}\sf320&\sf2\\\sf160&\sf2\\\sf80&\sf2\\\sf40&\sf2\\\sf20&\sf2\\\sf10&\sf2\\\sf5&\sf5\\\sf1\end{array}\\\sf 320=2^2\cdot2^2\cdot2^2\cdot5\\\begin{array}{c|c}\sf80&\sf2\\\sf40&\sf2\\\sf20&\sf2\\\sf10&\sf2\\\sf5&\sf5\\\sf1\end{array}\\\sf 80=2^2\cdot2^2\cdot5\end{array}}$

$\large\boxed{\begin{array}{l}\begin{array}{c|c}\sf800&\sf2\\\sf400&\sf2\\\sf200&\sf2\\\sf100&\sf2\\\sf50&\sf2\\\sf25&\sf5\\\sf5&\sf5\\\sf1\end{array}\\\sf 800=2^2\cdot2^2\cdot2\cdot5^2\end{array}}$

$\boxed{\begin{array}{l}\rm c)~\sf7\sqrt{450}-4\sqrt{320}+3\sqrt{80}-5\sqrt{800}-20\sqrt{5}\\\sf=7\sqrt{2\cdot3^2\cdot5^2}-4\sqrt{2^2\cdot2^2\cdot2^2\cdot5}+3\sqrt{2^2\cdot2^2\cdot5}\\\sf-5\sqrt{2^2\cdot2^2\cdot2\cdot5^2}-20\sqrt{5}\\\sf7\sqrt{450}-4\sqrt{320}+3\sqrt{80}-5\sqrt{800}-20\sqrt{5}\\\sf=7\cdot3\cdot5\sqrt{2}-4\cdot2\cdot2\cdot2\sqrt{5}+3\cdot2\cdot2\sqrt{5}-5\cdot2\cdot2\cdot5\sqrt{2}-20\sqrt{5}\end{array}}$

$\large\boxed{\begin{array}{l}\sf7\sqrt{450}-4\sqrt{320}+3\sqrt{80}-5\sqrt{800}-20\sqrt{5}\\\sf=105\sqrt{2}-32\sqrt{5}+12\sqrt{5}-100\sqrt{2}-20\sqrt{5}\\\sf=5\sqrt{2}-40\sqrt{5}\end{array}}$

$\Large\boxed{\begin{array}{l}\rm d)~\sf \sqrt{5}(\sqrt{3}-\sqrt{5}+\sqrt{8})\\\sf\sqrt{8}=\sqrt{2^2\cdot2}=2\sqrt{2}\\\sf\sqrt{5}\cdot(\sqrt{3}-\sqrt{5}+2\sqrt{2})\\\sf=\sqrt{5\cdot3}-\sqrt{5^2}+2\sqrt{5\cdot2}\\\sf=\sqrt{15}-5+2\sqrt{10}\end{array}}$