SOLUTION: Please help me simplify this equation: sqrt x^5 y^2

Question 424123: Please help me simplify this equation: sqrt x^5 y^2
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
First of all, equations have equals signs. What you have is called an expression. Second, I assume that the expression is:

and not

If so, then please use parentheses around radicands to make things clear. (Radicand the name for the expression inside a radical.) So your expression would be clear is you had posted:
sqrt(x^5y^2)

Simplifying a square root involves finding perfect square factors of the radicand (other than 1), if any. Your radicand has several perfect square factors so we will be able to simplify it. So we start by factoring the radicand so we can see the perfect square factors:

Next I like to use the Commutative Property to put the perfect square factors in the front. (You'll see why shortly.)

Next we use a property of radicals, , to split of the square root of a product into the product of the square roots of the factors:

The square roots of the perfect squares simplify:

This is the simplified expression.

Why the absolute value? And why is there an absolute value on the y but not on the ? The reasons:

Your original expression, , is a square root which must be non-negative (i.e. positive or zero).
The simplified expression we write at the end has to be just as non-negative as the original expression.
Since we have no idea what "y" is we do not know if it is positive, negative or zero. To ensure that our final expression is non-negative we put the absolute value on y.
We do not know what "x" is either. But we do not need an absolute value on because no matter what number x is cannot be negative.

Notes:

We used an absolute value on y because we had to ensure a non-negative value and because we didn't know what value y might have. However, some problems of this type include a statement saying something like: "Assume all variables have non-negative values." If there was such a statement (which you failed to include in your post):
- then the absolute value would no longer be necessary and you answer would be:
- the please include these statements in future postings.
The remaining square root is at the end of the term. This is the preferred (but not required) way to write a term with a square root. This is preferred because when the square root is at the end there is no possible confusion as to what part of the term is inside the square root and what is not. If the square root was at the front or in the middle of the term there might be confusion about this, especially when the term is hand-written. (This is why I put the perfect square factors in the front earlier.)

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