Video Transcript
Calculate eight plus the square root of negative 28 all squared multiplied by eight minus the square root of negative 28 squared.
We begin by recalling that if the square root of 𝑎 is equal to 𝑏, then the square root of negative 𝑎 is equal to 𝑏𝑖. This means that, in this question, we can rewrite the square root of negative 28 as the square root or root 28 multiplied by 𝑖. Our expression becomes eight plus root 28𝑖 all squared multiplied by eight minus root 28𝑖 squared. We can rewrite eight plus root 28𝑖 all squared as eight plus root 28𝑖 multiplied by eight plus root 28𝑖. The same method can be used to rewrite eight minus root 28𝑖 all squared.
We now have two pairs of parentheses written in the form 𝑎 plus 𝑏𝑖 multiplied by 𝑎 minus 𝑏𝑖. This is the factored form of the sum of two squares. 𝑎 squared plus 𝑏 squared is equal to 𝑎 plus 𝑏𝑖 multiplied by 𝑎 minus 𝑏𝑖. We can rewrite our expression as eight plus root 28𝑖 multiplied by eight minus root 28𝑖 all squared. Using the sum of two squares rule, the expression inside the square brackets simplifies to eight squared plus the square root of 28 squared.
We will then need to square this answer. Eight squared is equal to 64, and the square root of 28 squared is 28. As 64 plus 28 equals 92, our expression simplifies to 92 squared. This is equal to 8464. Eight plus the square root of negative 28 all squared multiplied by eight minus the square root of negative 28 all squared is equal to 8464.