question from Nova

**vasoolraja** · 04-07-2009, 03:27 PM

What is the solution of sqrt (z square - 4)square?

Nova solution is absolute value z square-4. How come the solution got absolute value?

**Kronos** · 04-07-2009, 03:58 PM

Because sqrt and square cancel each other out? Or am I misunderstanding your question.

**vasoolraja** · 04-07-2009, 04:04 PM

Kronos: Thanks for your answer. So when sqrt and square cancel each other, the result will have absolute symbol?

**Raison** · 04-07-2009, 04:18 PM

This is the rule you need to remember for problems like that:

rule
If n is an even positive integer, then

If n is an odd positive integer, then

**vasoolraja** · 04-07-2009, 04:28 PM

Originally Posted by Raison

This is the rule you need to remember for problems like that:

rule
If n is an even positive integer, then

If n is an odd positive integer, then

Thank you so much Raison!

**Raison** · 04-07-2009, 04:44 PM

You're welcome! I hope it helps!

**Kronos** · 04-07-2009, 08:22 PM

Originally Posted by Raison

This is the rule you need to remember for problems like that:

rule
If n is an even positive integer, then

If n is an odd positive integer, then

Would you mind shortly elaborating on the idea behind those formulas?

**Raison** · 04-08-2009, 06:41 AM

Those are the basic rules for finding the nth roots of perfect nth powers. It has been a while since I studied how they are derived, but they go back to the definition of the principal nth root of a real number.

The definition of the principal nth root of a real number states

ⁿ√a = b means bⁿ = a

If n is even, then a is nonnegative (a ≥ 0), and b is also nonnegative (b ≥ 0).

If n is odd, then a and b can be any real numbers.

For instance, if you have x = √9, and you’re solving for x, the answer is the nonnegative root, or 3; that is the principal root. (This is not the same as solving for x when x squared = 9, in which case your answer is +3, -3). We’re working with principal root in this problem.

To go back to the original problem being discussed, if you don’t put the absolute value symbol around the answer, that means that the expression under the radical could be either nonnegative or negative, which goes against the definition that says if n is even, then a can’t be negative.