Qual é a raiz do 4th do 80?

Responda:

$\sqrt[4]{80} = 2 \sqrt[4]{5}$ $root(4)(80) = 2root(4)(5)$

Explicação:

If $a, b, c > 0$ $a, b, c > 0$ então ${(a b)}^{c} = a^{c} b^{c}$ $(ab)^c = a^cb^c$

$\sqrt[4]{x} = x^{\frac{1}{4}}$ $root(4)(x) = x^(1/4)$ , assim $\sqrt[4]{a b} = \sqrt[4]{a} \sqrt[4]{b}$ $root(4)(ab) = root(4)(a)root(4)(b)$

If $a, b, c > 0$ $a, b, c > 0$ então ${(a^{b})}^{c} = a^{b c}$ $(a^b)^c = a^(bc)$

Assim:

$\sqrt[4]{a^{4}} = {(a^{4})}^{\frac{1}{4}} = a^{4 \cdot \frac{1}{4}} = a^{1} = a$ $root(4)(a^4) = (a^4)^(1/4) = a^(4*1/4) = a^1 = a$

Assim:

$\sqrt[4]{80} = \sqrt[4]{2^{4} \cdot 5} = \sqrt[4]{2^{4}} \sqrt[4]{5} = 2 \sqrt[4]{5}$ $root(4)(80) = root(4)(2^4*5) = root(4)(2^4)root(4)(5) = 2root(4)(5)$