1.- Aplica las propiedades de la radicación y la potenciación y simplifica a la mínima expresión
$$\displaystyle\left(\sqrt{a\sqrt[3]{8a^2}}\right)^{2}(2a)^{-4/3}$$→$\displaystyle\frac{a\sqrt[3]{8a^2}}{(2a)^{4/3}}$→$\displaystyle\frac{\sqrt[3]{2^3a^2a^3}}{\sqrt[3]{(2a)^4}}$→$\displaystyle\frac{2a\sqrt[3]{a^2}}{\sqrt[3]{2^4a^4}}$→$\displaystyle\frac{2a\sqrt[3]{a^2}}{2a\sqrt[3]{2a}}$→$\sqrt[3]{\displaystyle\frac{a^2}{2a}}$→$$→\sqrt[3]{\displaystyle\frac{a}{2}}\color{red}\bullet$$
2.- Desarrolla el siguiente Producto Notable
$$(a^{1/2}+b^{3/2})(a^{1/2}-b^{3/2})$$
→$\left(a^{1/2}\right)^2-\left(b^{3/2}\right)^2$→$\left(a^{1.2/2}\right)-\left(b^{3.2/2}\right)$→
$$a-b^3\color{red}\bullet$$
3.- Factorizar
$a→x^2-15x+50→(x-10)(x+5)\color{red}\bullet$
$b→a^{2n}-b^{2n}→\left(a^n\right)^2-\left(b^n\right)^2→(a^n+b^n)(a^n-b^n)$
$c→4x^2+8x-5→(2x)^2+4(2x)-5→$
$$→(2x+5)(2x-1)\color{red}\bullet$$
4.- Resolver, sin resolvente, excepto la nº c
$a→x^2-15x+50=0→(x-10)(x+5)=0$
$$→x-10=0→x=10\color{red}\bullet$$ $$→x+5=0→x=-5\color{red}\bullet$$
$b→4x^2+8x-5=0→(2x)^2+4(2x)-5=0→(2x+5)(2x-1)$
$$→2x+5=0→x=-\frac{5}{2}\color{red}\bullet$$ $$→2x-1=0→x=\frac{1}{2}\color{red}\bullet$$
$c→x^2-\sqrt{3}x-6=0$
$→x^2-\sqrt{3}x+\left(\sqrt{3}/2\right)^2=6+\left(\sqrt{3}/2\right)^2$
$→x^2-\sqrt{3}x+\left(x-\sqrt{3}/2\right)^2=6+\left(\sqrt{3}/2\right)^2$
$→\left(x-\sqrt{3}/2\right)^2=6+3/4$
$→\left(x-\sqrt{3}/2\right)^2=27/4→x-\sqrt{3}/2=\pm\sqrt{27/4}$
$→x=\sqrt{3}/2\pm3\sqrt{3}/2$
$$→x=\sqrt{3}/2+3\sqrt{3}/2→x=2\sqrt{3}\color{red}\bullet$$ $$→x=\sqrt{3}/2-3\sqrt{3}/2→x=-\sqrt{3}\color{red}\bullet$$
5.- Resolver el siguiente sistema
$\left \{ \begin{matrix} \displaystyle\frac{x-2y+3}{4}-\displaystyle\frac{2x-y}{12}=-\displaystyle\frac{7}{6}
\\ \displaystyle\frac{x}{3}-y=-5 \end{matrix}\right.$ → $\left \{ \begin{matrix} 3x-6y+9-2x+y=-14
\\ x-3y=-15 \end{matrix}\right.$
→ $-2y=-8y$ → $y=4\color{red}\bullet$ → $x=-3\color{red}\bullet$
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