Question

Considerando que x² = 5³, y³ = 5⁴ e z⁶ = 5², entao o valor n que satisfaz a igualdade 5n = (x.y.z)²⁴ é

Answer 1

Olá!

Explicação passo a passo:

Note que,

$\begin{cases} \mathtt{x^2 = 5^3 \Rightarrow (x^2)^{1/2} = (5^3)^{1/2} \Rightarrow x = 5^{5/2}} \\ \mathtt{y^3 = 5^4 \Rightarrow (y^3)^{1/3} = (5^4)^{1/3} \Rightarrow y = 5^{4/3}} \\ \mathtt{z^6 = 5^2 \Rightarrow (z^6)^{1/6} = (5^2)^{1/6} \Rightarrow z = 5^{2/6}}\end{cases}$

Isto posto,

$\\ \mathtt{5n = (x \cdot y \cdot z)^{24}} \\\\ \mathtt{5n = \left ( 5^{5/2} \cdot 5^{4/3} \cdot 5^{2/6} \right )^{24}} \\\\ \mathtt{5n = 5^{120/2} \cdot 5^{96/3} \cdot 5^{48/6}} \\\\ \mathtt{5n = 5^{60} \cdot 5^{32} \cdot 5^{8}} \\\\ \mathtt{5n = 5^{60 + 32 + 8}} \\\\ \mathtt{5n = 5^{100}} \\\\ \mathtt{n = 5^{100 - 1}} \\\\ \boxed{\boxed{\mathtt{n = 5^{99}}}}$