SOLUTION: Simplify the following expression. square root of -6 and -42. Both numbers are next to each other with the square root sign in front of each number.

Algebra ->  Square-cubic-other-roots -> SOLUTION: Simplify the following expression. square root of -6 and -42. Both numbers are next to each other with the square root sign in front of each number.      Log On


   

Question 609671: Simplify the following expression.
square root of -6 and -42. Both numbers are next to each other with the square root sign in front of each number.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%28-6%29%2Asqrt%28-42%29
When the radicand (the number inside) of a square root is negative, then the number is what is called an imaginary number. When working with imaginary numbers the first thing to do is to rewrite them in terms of "i":
i+=+sqrt%28-1%29 and i%5E2+=+-1

Rewriting your numbers in terms of i. Start by factoring out -1 in each radicand:
sqrt%28-1%2A6%29%2Asqrt%28-1%2A42%29
Use a property of radicals, root%28a%2C+p%2Aq%29+=+root%28a%2C+p%29%2Aroot%28a%2C+q%29, to split the factors into separate square roots:
sqrt%28-1%29%2Asqrt%286%29%2Asqrt%28-1%29%2Asqrt%2842%29
Replace sqrt%28-1%29 with "i":
i%2Asqrt%286%29%2Ai%2Asqrt%2842%29

Now we can proceed with "normal" simplification. Using the Commutative and Associative Properties to rearrange the order and the grouping:
%28i%2Ai%29%2A%28sqrt%286%29%2Asqrt%2842%29%29
Multiplying:
%28i%5E2%29%2A%28sqrt%28252%29%29
Replacing i%5E2 with -1:
%28-1%29%2A%28sqrt%28252%29%29
Simplifying the square root:
%28-1%29%2A%28sqrt%2836%2A7%29%29
%28-1%29%2A%28sqrt%2836%29%2Asqrt%287%29%29
%28-1%29%2A%286%2Asqrt%287%29%29
-6%2Asqrt%287%29
This is the simplified expression.