Simplifica la siguiente expresión: $$ \left(\frac{2}{3}\right)^{5}\left(\frac{2}{3}\right)^{0}\left(\frac{2}{3}\right)^{-3}\left(\frac{81}{16}\right)^{-2} $$ $$ \left(\frac{3}{2}\right)^{-5}\left(\frac{2}{3}\right)\left[\left(\frac{2}{3}\right)^{5}\right]^{2}\left(\frac{8}{27}\right)^{3} $$

Question

UpStudy AI · Accepted Answer

1. Empezamos simplificando

$$
\left(\frac{2}{3}\right)^{5}\left(\frac{2}{3}\right)^{0}\left(\frac{2}{3}\right)^{-3}\left(\frac{81}{16}\right)^{-2}
$$

a) Combinamos las potencias con base $$\frac{2}{3}$$:

$$
\left(\frac{2}{3}\right)^{5+0+(-3)} = \left(\frac{2}{3}\right)^2
$$

b) Reescribimos la última potencia:

$$
\left(\frac{81}{16}\right)^{-2} = \left(\frac{16}{81}\right)^2
$$

Recordamos que $$16=2^4$$ y $$81=3^4$$, así que

$$
\left(\frac{16}{81}\right)^2 = \left(\frac{2^4}{3^4}\right)^2 = \frac{2^8}{3^8} = \left(\frac{2}{3}\right)^8
$$

c) Multiplicamos las dos componentes:

$$
\left(\frac{2}{3}\right)^2 \cdot \left(\frac{2}{3}\right)^8 = \left(\frac{2}{3}\right)^{2+8} = \left(\frac{2}{3}\right)^{10}
$$

2. Ahora simplificamos

$$
\left(\frac{3}{2}\right)^{-5}\left(\frac{2}{3}\right)\left[\left(\frac{2}{3}\right)^{5}\right]^{2}\left(\frac{8}{27}\right)^{3}
$$

a) Convertimos la primera potencia:

$$
\left(\frac{3}{2}\right)^{-5} = \left(\frac{2}{3}\right)^5
$$

b) La expresión queda:

$$
\left(\frac{2}{3}\right)^5 \cdot \left(\frac{2}{3}\right) \cdot \left[\left(\frac{2}{3}\right)^{5}\right]^2 \cdot \left(\frac{8}{27}\right)^{3}
$$

Notamos que:

$$
\left[\left(\frac{2}{3}\right)^5\right]^2 = \left(\frac{2}{3}\right)^{10}
$$

c) Combinamos las potencias de $$\frac{2}{3}$$:

$$
\left(\frac{2}{3}\right)^5 \cdot \left(\frac{2}{3}\right)^1 \cdot \left(\frac{2}{3}\right)^{10} = \left(\frac{2}{3}\right)^{5+1+10} = \left(\frac{2}{3}\right)^{16}
$$

d) Simplificamos la última potencia:

Observamos que

$$
\frac{8}{27} = \frac{2^3}{3^3} = \left(\frac{2}{3}\right)^3
$$

Entonces,

$$
\left(\frac{8}{27}\right)^3 = \left[\left(\frac{2}{3}\right)^3\right]^3 = \left(\frac{2}{3}\right)^9
$$

e) Multiplicamos:

$$
\left(\frac{2}{3}\right)^{16} \cdot \left(\frac{2}{3}\right)^9 = \left(\frac{2}{3}\right)^{16+9} = \left(\frac{2}{3}\right)^{25}
$$

Respuestas finales:

1. $$\left(\frac{2}{3}\right)^{10}$$

2. $$\left(\frac{2}{3}\right)^{25}$$
$$ \left(\frac{2}{3}\right)^{10} \quad \text{y} \quad \left(\frac{2}{3}\right)^{25} $$

Simplifica la siguiente expresión:

Upstudy AI Solution

Answer

Solution

Mind Expander

Related Questions

Latest Algebra Questions