Video Transcript
Simplify the square root of 17 minus the square root of negative 11 multiplied by the square root of 11 minus the square root of negative 17.
We begin by recalling how we can rewrite the square root of a negative number. If the square root of the positive number 𝑎 is equal to 𝑏, then the square root of negative 𝑎 is equal to 𝑏𝑖. This means that the square root of negative 11 can be rewritten as the square root of 11 multiplied by 𝑖. In the same way, the square root of negative 17 is equal to root 17 𝑖. The expression in this question can therefore be rewritten as root 17 minus root 11 𝑖 multiplied by root 11 minus root 17 𝑖.
We can now expand the brackets or distribute the parentheses using the rules that the square root of 𝑎 multiplied by the square root of 𝑏 is equal to the square root of 𝑎 multiplied by 𝑏 and also that the square root of 𝑎 squared is equal to 𝑎. We will use the FOIL method.
And multiplying the first terms, root 17 and root 11, gives us root 187. This is because 17 multiplied by 11 is 187. Multiplying the outer terms gives us negative 17𝑖 as root 17 multiplied by root 17 is 17. Multiplying the inner terms gives us negative 11𝑖. Finally, multiplying the last terms gives us root 187 𝑖 squared.
From our knowledge of imaginary numbers, we recall that 𝑖 squared equals negative one. This means that we have root 187 minus root 187, so these terms will cancel. Negative 17 minus 11 is equal to negative 28. Therefore, our expression simplifies to negative 28𝑖.
The square root of 17 minus the square root of negative 11 multiplied by the square root of 11 minus the square root of negative 17 is equal to negative 28𝑖.
