SOLUTION: Simplify {{{ sqrt (3x^4v^3) sqrt (9x^3v^3)}}} Assume that all variables represent positive real numbers.

Question 392087: Simplify

Assume that all variables represent positive real numbers.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

With an expression like this you can either:
- Multiply and then simplify;
or
- Simplify each square root, then multiply and then simplify again.

The second approach may seem like extra work because it is three steps. But when the radicands (the expressions inside a radical) are more complex (e.g. large coefficients, multiple terms) the second approach is actually easier.

Your radicands are fairly simple so we will use the first approach. To multiply radicals we use the property :

Inside the square root it is all multiplication. So we can use the Commutative and Associative Properties to rearrange the order and grouping in any way we choose. So we can multiply the coefficients, the factors with "x" and the factors with "v" separately giving us:

The multiplying is finished. Now we simplify. Simplifying square roots involves finding perfect square factors, if any, of the radicand.

For reasons I will explain shortly, I like to use the Commutative Property to rearrange the order so that all the perfect square factors are in front:

Now we use the property again. This time, however, we will use it from right to left so that we can put all the perfect square factors into their own square root. (Put all the factors that are not perfect squares into one square root.):

All the square roots of the perfect squares will simplify:

As you can see, the parts of the expression that are not inside a square root are all in front of the remaining square root. This is where we want them and this is why I put the perfect square factors in front earlier. The expression will now simplify to:

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