Question

for the values listed in the table, i represents the imaginary unit. select all the cells in the table for which the product of the row value and the column value is -1. value i i² i³ i⁴ i⁵ i⁶

Explanation:

Step1: Recall the properties of \(i\)

We know that \(i=\sqrt{- 1}\), \(i^{2}=-1\), \(i^{3}=i^{2}\times i=-i\), \(i^{4}=(i^{2})^{2}=(-1)^{2} = 1\), \(i^{5}=i^{4}\times i=i\), \(i^{6}=i^{4}\times i^{2}=1\times(-1)=-1\)

Step2: Calculate the product of row - column values

• For the cell with row \(i\) and column \(i^{4}\): \(i\times i^{4}=i\times1 = i

eq - 1\)

• For the cell with row \(i\) and column \(i^{5}\): \(i\times i^{5}=i\times i=i^{2}=-1\)
• For the cell with row \(i\) and column \(i^{6}\): \(i\times i^{6}=i\times(-1)=-i

eq - 1\)

• For the cell with row \(i^{2}\) and column \(i^{4}\): \(i^{2}\times i^{4}=(-1)\times1=-1\)
• For the cell with row \(i^{2}\) and column \(i^{5}\): \(i^{2}\times i^{5}=(-1)\times i=-i

eq - 1\)

• For the cell with row \(i^{2}\) and column \(i^{6}\): \(i^{2}\times i^{6}=(-1)\times(-1)=1

eq - 1\)

• For the cell with row \(i^{3}\) and column \(i^{4}\): \(i^{3}\times i^{4}=(-i)\times1=-i

eq - 1\)

• For the cell with row \(i^{3}\) and column \(i^{5}\): \(i^{3}\times i^{5}=(-i)\times i=-i^{2}=1

eq - 1\)

• For the cell with row \(i^{3}\) and column \(i^{6}\): \(i^{3}\times i^{6}=(-i)\times(-1)=i

eq - 1\)

Answer:

The cells corresponding to \(i\times i^{5}\) and \(i^{2}\times i^{4}\)