If 𝑟 is equal to two plus 𝑖, express 𝑟 cubed in the form 𝑎 plus 𝑏𝑖.
We are asked to express 𝑟 cubed in the form 𝑎 plus 𝑏𝑖. Therefore, we need to cube two plus 𝑖. This can be rewritten as two plus 𝑖 multiplied by two plus 𝑖 multiplied by two plus 𝑖. We will begin by distributing the first two parentheses, otherwise known as expanding the brackets using the FOIL method. Multiplying the first terms gives us four. The outer terms multiply to give us two 𝑖. The same is true of the inner terms. Finally, the last terms have a product of 𝑖 squared. Two plus 𝑖 multiplied by two plus 𝑖 is equal to four plus two 𝑖 plus two 𝑖 plus 𝑖 squared.
Before multiplying this by two plus 𝑖, we will simplify by collecting like terms. Two 𝑖 plus two 𝑖 is equal to four 𝑖. From our knowledge of imaginary numbers, we know that 𝑖 squared is equal to negative one. This means that four plus two 𝑖 plus two 𝑖 plus 𝑖 squared simplifies to three plus four 𝑖. We will now multiply this expression by two plus 𝑖 using the FOIL method once again. Multiplying the first terms gives us six. Three multiplied by 𝑖 is three 𝑖. Four 𝑖 multiplied by two is equal to eight 𝑖. Finally, four 𝑖 multiplied by 𝑖 is four 𝑖 squared.
Our expression simplifies to six plus three 𝑖 plus eight 𝑖 plus four 𝑖 squared. Once again, we can use the fact that 𝑖 squared equals negative one. Therefore, four 𝑖 squared is negative four. Six minus four is equal to two, and three 𝑖 plus eight 𝑖 is equal to 11𝑖. If 𝑟 is equal to two plus 𝑖, then 𝑟 cubed written in the form 𝑎 plus 𝑏𝑖 is two plus 11𝑖.