Demostrar

Question

Mi profesor de matemáticas dejó dos preguntas la semana pasada, probar (1) <span class="math-container">$6^9>10^7$</span> y (2) <span class="math-container">$7^{71}>75^{32}.$</span>

Hice la primera pregunta: <span class="math-container">\begin{align}\frac{6^9}{10^7}&=\frac{4}{5}\times\frac{27^3}{25^3}\&=0.8\times1.08^3\&>0.8\times(1+3\times0.08+3\times0.08^2)\&>0.8\times(1+3\times0.086)\&>0.8\times1.25=1.\end{align}</span>

Pero no puedo resolver el segundo, lo calculé en mi computadora, <span class="math-container">$\frac{7^{71}}{75^{32}}=1.000000949\cdots$</span>

Answer 1

Partiendo de la desigualdad en cuestión y transformándola a una desigualdad manifiestamente verdadera por pasos reversibles, la desigualdad en cuestión demostrará la desigualdad en cuestión. <span class="math-container">$$7^{71}>75^{32}?$$<span class="math-container">$$7^{71}>5^{32}\times 15^{32}?$$</span><span class="math-container">$$2^{32}\times 7^{71}>15^{32}\times 10^{32}?$$</span><span class="math-container">$$7^{71}>(7.5)^{32}\times 10^{32}?$$</span><span class="math-container">$$7^{39}>\left(\frac{7.5}{7}\right)^{32}\times 10^{32}?$$</span><span class="math-container">$$\frac{7.5}{7}<span class="math-container">$$\left(\frac{7.5}{7}\right)^{32}<span class="math-container">$$\left(\frac{7.5}{7}\right)^{32}\times 10^{32}<span class="math-container">$$7^{39}>9.09543680\times 10^{32}$$</span></span></span></span></span>

Sí, <span class="math-container">$7^{39}>\left(\frac{7.5}{7}\right)^{32}\times 10^{32},$</span> por lo tanto <span class="math-container">$7^{71}>75^{32}.$</span>