### Video Transcript

What is the value of 𝑖 to the seventh power?

We recall that an imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit 𝑖. This is defined by the property 𝑖 squared is equal to negative one. 𝑖 cubed is equal to 𝑖 squared multiplied by 𝑖. As 𝑖 squared is equal to negative one, 𝑖 cubed is equal to negative one multiplied by 𝑖. This simplifies to negative 𝑖. We can repeat this process to find an expression for 𝑖 to the fourth power. As 𝑖 cubed is equal to negative 𝑖, 𝑖 to the fourth power is equal to negative 𝑖 multiplied by 𝑖. This is equal to negative 𝑖 squared, which in turn is equal to negative negative one, or positive one.

We can then continue this pattern to calculate 𝑖 to the fifth power. Multiplying 𝑖 to the fourth power by 𝑖 is the same as multiplying one by 𝑖. This means that 𝑖 to the fifth power is simply equal to 𝑖. 𝑖 to the sixth power is equal to 𝑖 to the fifth power multiplied by 𝑖. This gives us 𝑖 multiplied by 𝑖, which is 𝑖 squared, and we know this is equal to negative one. Finally, we have 𝑖 to the seventh power, which is equal to 𝑖 to the sixth power multiplied by 𝑖. This in turn is equal to negative one multiplied by 𝑖, or negative 𝑖. The value of 𝑖 to the seventh power is negative 𝑖.

We notice that when our power is odd, we have an imaginary answer, a value involving 𝑖. When the power of 𝑖 on the other hand is even, we have a real answer, either one or negative one. 𝑖 raised to any power will be equal to one, negative one, 𝑖, or negative 𝑖. It will be equal to one or negative one when the power is even and equal to 𝑖 or negative 𝑖 when the power is odd.