SOLUTION: Multiply and simplify the square root of 2b^7 by the square root of 10c^8 by factoring and assuming that all expressions under radicals represent nonnegative numbers?

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Question 401450: Multiply and simplify the square root of 2b^7 by the square root of 10c^8 by factoring and assuming that all expressions under radicals represent nonnegative numbers?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%282b%5E7%29%2Asqrt%2810c%5E8%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%282b%5E7%2A10c%5E8%29
which simplifies to
sqrt%2820b%5E7%2Ac%5E8%29
We can simplify this further if we can find perfect square factors in the radicand (the expression within a radical). 20 is not a perfect square but 4 is a perfect square and it is a factor of 20. b%5E7 is not a perfect square not because 7 is not a perfect square but because it is not divisible by 2. But b%5E7+=+b%5E6%2Ab and b%5E6 is a perfect square not because 6 is a perfect square (which it is not) but because 6 is divisible by 2. c%5E8 is itself a perfect square not because 8 is a perfect square (which it is not) but because 8 is divisible by 2. So we can factor the radicand as follows
sqrt%284%2A5%2Ab%5E6%2Ab%2Ac%5E8%29
For reasons you will see shortly, I like to use the Commutative Property to change the order of the factors so that the perfect squares are in front:
sqrt%284%2Ab%5E6%2Ac%5E8%2A5%2Ab%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%284%29%2Asqrt%28b%5E6%29%2Asqrt%28c%5E8%29%2Asqrt%285%2Ab%29
The square roots of the perfect squares simplify:
sqrt%282%5E2%29%2Asqrt%28%28b%5E3%29%5E2%29%2Asqrt%28%28c%5E4%29%5E2%29%2Asqrt%285%2Ab%29
2%2Ab%5E3%2Ac%5E4%2Asqrt%285%2Ab%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.

The part about "non-negative" was useful in this problem. Even-numbered roots (square roots, 4th roots, 6th roots, etc.) are supposed to be non-negative (unless there is a minus in front). So when you simplify even-numbered roots, the simplified expression must be just as non-negative as the original expression. For this reason absolute value is sometimes used to ensure that the simplified expression is just as non-negative. But since we were told that the expressions were non-negative there was no possibility that our simplified expression, which involves the non-negative variables, could ever be negative.