The original can be rewritten as [tex] \sqrt{-1*6}* \sqrt{-1*24} [/tex]. Because i^2 is equal to -1, we can replace the -1 in each radicand with i^2, like this: [tex] \sqrt{i^2*6}* \sqrt{i^2*24} [/tex]. Now, i-squared is a perfect square that can be pulled out of each radicand as a single i. [tex]i \sqrt{6}*i \sqrt{24} [/tex]. 24 has a perfect square hidden in it. 4 * 6 = 24 and 4 is a perfect square. So let's break this up, step by step. [tex]i \sqrt{6}*i \sqrt{4*6} [/tex] and then [tex]i \sqrt{6}*2i \sqrt{6} [/tex]. We will now multiply the i and the 2i, and multiply the square root of 6 times the square root of 6: [tex]2i^2 \sqrt{36} [/tex]. 36 itself is a perfect square because 6 * 6 = 36. So we will do that simplification now. [tex]2i^2(6)[/tex]. Multiplying the 2 and the 6 gives us [tex]12i^2[/tex]. But here we are back to the fact that i-squared is equal to -1, so 2(-1)(6) = -12. See how that works?
