SOLUTION: Express the radical in simplified form. Assume all variables represent positive real numbers. square root of ((72a^4)(b^3))

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Question 274469: Express the radical in simplified form. Assume all variables represent positive real numbers.
square root of ((72a^4)(b^3))
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
sqrt%28%2872a%5E4%29%28b%5E3%29%29
Simplifying square roots is just a matter of
  • "reducing" the square root by finding perfect square factors, if any, and...
  • eliminating square roots in denominators, if any

Since you have no denominators we only have to deal with the first item: Finding perfect square factors, if any. There are several:
sqrt%2836%2A2%2A%28a%5E2%29%5E2%2Ab%5E2%2Ab%29%29
Now we use a property of radicals, root%28a%2C+p%2Aq%29+=+root%28a%2C+p%29%2Aroot%28a%2C+q%29 to separate all the perfect square factors into their own square roots:
sqrt%2836%29%2Asqrt%282%29%2Asqrt%28%28a%5E2%29%5E2%29%2Asqrt%28b%5E2%29%2Asqrt%28b%29
We can now simplify the square roots of the perfect squares:
6%2Asqrt%282%29%2Aa%5E2%2Ab%2Asqrt%28b%29
Since this is all multiplication we can use the Commutative Property to rearrange the factors. We'll put the square roots in the back:
6a%5E2b%2Asqrt%282%29%2Asqrt%28b%29
And last of all we can use the same property as before but this time we'll use it to combine the remaining square roots:
6a%5E2b%2Asqrt%282b%29
This is the simplified form of your original expression.