Can somebody help me understand imaginary numbers???how do I simplify {i}^(13)how do I solve √( - 49)

Question

can somebody help me understand imaginary numbers???how do I simplify
`{i}^(13)`how do I solve
`√( - 49)`

Answer 1

`i^(13)`

`\begin{gathered} i=\sqrt[]{-1} \\ So \\ i^2=-1 \end{gathered}`
`\begin{gathered} \text{Express:} \\ i^(13) \\ As \\ i^(13)=i^2\cdot i^2\cdot i^2\cdot i^2\cdot i^2\cdot i^2\cdot i \end{gathered}`
`\begin{gathered} i^(13)=(-1)\cdot(-1)\cdot(-1)\cdot(-1)\cdot(-1)\cdot(-1)\cdot i \\ i^(13)=1\cdot i \\ i^(13)=i \end{gathered}`

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`\begin{gathered} \sqrt[]{-49} \\ \text{Use this property:} \\ \sqrt[]{a\cdot b}=\sqrt[]{a}\cdot\sqrt[]{b} \\ So\colon \\ \sqrt[]{-49}=\sqrt[]{49\cdot(-1)}=\sqrt[]{49}\cdot\sqrt[]{-1} \\ \text{Where:} \\ \sqrt[]{49}=7 \\ \sqrt[]{-1}=i \\ so\colon \\ \sqrt[]{49}\cdot\sqrt[]{-1}=7i \end{gathered}`