SOLUTION: Multiply and simplify the square root of 20 by the square root of 40 by factoring?

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Question 401451: Multiply and simplify the square root of 20 by the square root of 40 by factoring?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%2820%29%2Asqrt%2840%29
When multiplying radicals you have to choice of
  • Simplify, multiply and simplify again; or
  • Multiply and simplify

The first approach seems like more work. But when the multiplication is complicated it can help a lot if you have simplified first. Your multiplication is not very complicated so we will use the second approach.

Multiplying radicals uses a basic property of radicals: root%28a%2C+p%29%2Aroot%28a%2C+q%29+=+root%28a%2C+p%2Aq%29. Using this to multiply your radicals we get:
sqrt%2820%2A40%29
which simplifies to
sqrt%28800%29
We can simplify this further if we can find perfect square factors in the radicand (the expression within a radical). 800 is a fairly large number and it will have many factors. And some of these factors will be perfect squares. The fastest way to simplify is to find the largest perfect square factor. 400 is 20%5E2 an dit is a factor of 800. So we can factor the radicand as follows. (For reasons you will see shortly, I put the perfect square factor front.)
sqrt%28400%2A2%29
Now we use the same property as earlier. But this time we are using it in the opposite direction to take a single sqaure root of a product and break it into a product of the square roots of each factor:
sqrt%28400%29%2Asqrt%282%29
The square root of the perfect square simplify:
sqrt%2820%5E2%29%2Asqrt%282%29
20%2Asqrt%282%29
Note how the radical is at the end. This is the normal way to write terms like this. The radical is at the end now because we put the perfect square factors in front earlier.