Комплексни бројеви 3

Текст задатака објашњених у видео лекцији.

Пр.1   Одредити збир и разлику датих комплексних бројева.

           ${z_1} = 5 - 3i$   ${z_2} =  - 4 + 7i$      ${z_1} + {z_2} = ?$  ${z_1} - {z_2} = ?$

Пр.2   Одредити збир и разлику датих комплексних бројева.

           ${z_1} = 1 - 2\sqrt 3  + 4i$   ${z_2} = \sqrt 3  - 5 - 2\sqrt 3 i$   

           ${z_1} + {z_2} = ?$   ${z_1} -  {z_2} = ?$   

Пр.3   Израчунати вредност израза.

           $\sqrt { - 75}  + 2\sqrt { - 108}  - i\sqrt { - 12}  + 3\sqrt {48}  = $

Пр.4   Израчунати вредност израза.

           $\sqrt[3]{{ - 8}} - 2\sqrt { - 0,04}  + \sqrt[3]{{ - 0,064}} + \sqrt { - 81}  = $

Пр.1

\[\begin{gathered}
{z_1} = {\text{ }}5{\text{ }} - {\text{ }}3i \hfill \\
{z_2} = {\text{ }} - {\text{ }}4{\text{ }} + {\text{ }}7i \hfill \\
{z_1} + {\text{ }}{z_2}{\text{ = }}5{\text{ }} - {\text{ }}3i - {\text{ }}4{\text{ }} + {\text{ }}7i = 1 + 4i \hfill \\
{z_1}{\text{ - }}{z_2}{\text{ = }}5{\text{ }} - {\text{ }}3i - \left( { - {\text{ }}4{\text{ }} + {\text{ }}7i} \right) = 5{\text{ }} - {\text{ }}3i + {\text{ }}4{\text{ }} - {\text{ }}7i = 9 - 10i \hfill \\
\end{gathered} \]

Пр. 2

\[\begin{gathered}
{z_1} = {\text{ 1 }} - {\text{ 2}}\sqrt 3 + 4i \hfill \\
{z_2} = {\text{ }}\sqrt 3 {\text{ - 5 - 2}}\sqrt 3 {\text{ i}} \hfill \\
{z_1} + {\text{ }}{z_2}{\text{ = 1 }} - {\text{ 2}}\sqrt 3 + 4i{\text{ }} + {\text{ }}\sqrt 3 {\text{ - 5 - 2}}\sqrt 3 {\text{ i}} = - 4 - \sqrt 3 - 4i - 2\sqrt 3 i = \hfill \\
= \left( { - 4 - \sqrt 3 } \right) + \left( {4 - 2\sqrt 3 } \right)i \hfill \\
\end{gathered} \]

\[\begin{gathered}
{z_1} - {z_2} = 1 - 2\sqrt 3 + 4i - \left( {\sqrt 3 - 5 - 2\sqrt 3 i} \right) = \hfill \\
= 1 - 2\sqrt 3 + 4i - \sqrt 3 + 5 + 2\sqrt 3 i = \hfill \\
= 6 - 3\sqrt 3 + 4i + 2\sqrt 3 i = \left( {6 - 3\sqrt 3 } \right) + \left( {4 + 2\sqrt 3 } \right)i \hfill \\
\end{gathered} \]

Пр. 3

\[\begin{gathered}
\sqrt { - 75} + 2\sqrt { - 108} - i\sqrt { - 12} + 3\sqrt {48} = \hfill \\
= \sqrt { - 1 \cdot 25 \cdot 3} + 2\sqrt { - 1 \cdot 36 \cdot 3} - i\sqrt { - 1 \cdot 4 \cdot 3} + 3\sqrt {3 \cdot 16} = \hfill \\
= \sqrt { - 1} \sqrt {25} \sqrt 3 + 2\sqrt { - 1} \sqrt {36} \sqrt 3 - i\sqrt { - 1} \sqrt 4 \sqrt 3 + 3\sqrt 3 \sqrt {16} = \hfill \\
= i5\sqrt 3 + 2i6\sqrt 3 - {i^2}2\sqrt 3 + 3\sqrt 3 \cdot 4 = \hfill \\
= 5\sqrt 3 i + 12\sqrt 3 i + 2\sqrt 3 + 12\sqrt 3 = 17\sqrt 3 i + 14\sqrt 3 \hfill \\
\end{gathered} \]

Пр. 4

\[\begin{gathered}
\sqrt[3]{{ - 8}} - 2\sqrt { - 0,04} + \sqrt[3]{{ - 0,064}} + \sqrt { - 81} = \hfill \\
= \sqrt[3]{{{{\left( { - 2} \right)}^3}}} - 2i\sqrt {{{\left( {0,2} \right)}^2}} + \sqrt[3]{{{{\left( { - 0,4} \right)}^3}}} + i\sqrt {81} = \hfill \\
= - 2 - 2i \cdot 0,2 + \left( { - 0,4} \right) + i \cdot 9 = - 2 - 0,4i - 0,4 + 9i = - 2,4 + 8,6i \hfill \\
\end{gathered} \]