Question

which of the following options is equivalent to $z_1cdot z_2$? select all that apply.
$z_1 = \sqrt{3}-\mathrm{i}$
$z_2 = 4\sqrt{6}+5\mathrm{i}$
$\square z_1cdot z_2 = 12\sqrt{2}-4\mathrm{i}\sqrt{6}+5\mathrm{i}\sqrt{3}-5\mathrm{i}$
$\square z_1cdot z_2 = 4\sqrt{18}-20\mathrm{i}\sqrt{18}-5\mathrm{i}^2$
$\square z_1cdot z_2 = 12\sqrt{2}-4\mathrm{i}\sqrt{6}+5\mathrm{i}\sqrt{3}+5$
$\square z_1cdot z_2 = 4\sqrt{18}-4\mathrm{i}\sqrt{6}+5\mathrm{i}\sqrt{3}-5\mathrm{i}^2$
$\square z_1cdot z_2 = 4\sqrt{18}+\mathrm{i}\sqrt{9}-5\mathrm{i}^2$
$\square z_1.z_2 = 12\sqrt{2}-4\mathrm{i}\sqrt{6}+5\mathrm{i}\sqrt{3}-5(-1)$

Explanation:

Explicación:

Paso 1: Multiplicar complejos

Multiplicamos $z_1=\sqrt{3}-i$ y $z_2 = 4\sqrt{6}+5i$ usando la regla $(a + bi)(c+di)=ac + adi + bci+bdi^{2}$.
$$(\sqrt{3}-i)(4\sqrt{6}+5i)=\sqrt{3}\times4\sqrt{6}+\sqrt{3}\times5i - i\times4\sqrt{6}-i\times5i$$

Paso 2: Simplificar términos

Simplificamos cada término:

• $\sqrt{3}\times4\sqrt{6}=4\sqrt{18}=4\times3\sqrt{2}=12\sqrt{2}$.
• $\sqrt{3}\times5i = 5i\sqrt{3}$.
• $-i\times4\sqrt{6}=-4i\sqrt{6}$.
• $-i\times5i=- 5i^{2}$, y como $i^{2}=-1$, entonces $-5i^{2}=5$.
• Entonces $(\sqrt{3}-i)(4\sqrt{6}+5i)=12\sqrt{2}-4i\sqrt{6}+5i\sqrt{3}+5$.

Respuesta:

$z_1\cdot z_2 = 12\sqrt{2}-4i\sqrt{6}+5i\sqrt{3}+5$
$z_1\cdot z_2 = 12\sqrt{2}-4i\sqrt{6}+5i\sqrt{3}-5(-1)$
$z_1\cdot z_2 = 4\sqrt{18}-4i\sqrt{6}+5i\sqrt{3}-5i^{2}$

Answer:

Explicación:

Paso 1: Multiplicar complejos

Multiplicamos $z_1=\sqrt{3}-i$ y $z_2 = 4\sqrt{6}+5i$ usando la regla $(a + bi)(c+di)=ac + adi + bci+bdi^{2}$.
$$(\sqrt{3}-i)(4\sqrt{6}+5i)=\sqrt{3}\times4\sqrt{6}+\sqrt{3}\times5i - i\times4\sqrt{6}-i\times5i$$

Paso 2: Simplificar términos

Simplificamos cada término:

• $\sqrt{3}\times4\sqrt{6}=4\sqrt{18}=4\times3\sqrt{2}=12\sqrt{2}$.
• $\sqrt{3}\times5i = 5i\sqrt{3}$.
• $-i\times4\sqrt{6}=-4i\sqrt{6}$.
• $-i\times5i=- 5i^{2}$, y como $i^{2}=-1$, entonces $-5i^{2}=5$.
• Entonces $(\sqrt{3}-i)(4\sqrt{6}+5i)=12\sqrt{2}-4i\sqrt{6}+5i\sqrt{3}+5$.

Respuesta:

$z_1\cdot z_2 = 12\sqrt{2}-4i\sqrt{6}+5i\sqrt{3}+5$
$z_1\cdot z_2 = 12\sqrt{2}-4i\sqrt{6}+5i\sqrt{3}-5(-1)$
$z_1\cdot z_2 = 4\sqrt{18}-4i\sqrt{6}+5i\sqrt{3}-5i^{2}$