# Simplifying Absolute Values

1. Oct 17, 2012

### Rably

1. The problem statement, all variables and given/known data
Simplify.

a) $$\sqrt{x^6}$$
b) $$8 \sqrt{x^7y^{10}} - 10 \sqrt{x^7y^{10}}$$

For b, it's y^10. I can't make it look right for some reason.
Mod note: Fixed the exponent.

2. Relevant equations

3. The attempt at a solution
I can simplify all of them but I don't know when or where I need to put in absolute value symbols to the solution. I know the solution because my book shows me, but I don't understand why the absolute values are where they are.

For instance, the solution to a is $$|x^3|$$ but I can only get to $$x^3$$ without becoming confused.
For b, i can get to $$-2x^3y^5 \sqrt{x}$$ but the solution is $$-2x^3|y^5| \sqrt{x}$$
So how do you know when an absolute value is required?

Last edited by a moderator: Oct 17, 2012
2. Oct 17, 2012

### SammyS

Staff Emeritus
Use y^{10}
$\displaystyle \sqrt{u^2}=|u|$

Also, remember that if n is a positive integer, then $u^{2n}\ge0\,,$ so there is no need to use absolute value.

3. Oct 17, 2012

### Rably

So in my scenario, x has to be positive because it's still underneath the square root so I don't need to put an absolute value around x^3, whereas y can be either positive or negative because it's not still beneath the square root so an absolute value is required?

4. Oct 17, 2012

### Staff: Mentor

No. x can be any real number.

Like SammyS said,
$$\sqrt{x^2} = |x|$$

so
$$\sqrt{x^6} = \sqrt{(x^3)^2} = |x^3|$$

This is also the same as |x|3.

5. Oct 17, 2012

### SammyS

Staff Emeritus
Which scenario ?

If x is to an even power, that result is non-negative. So that can be under a radical -- actually one signifying a square root -- no matter what value x has.

6. Oct 17, 2012

### Rably

I was referring to question b. Do you mean if x has an even power in the initial equation or in the solution?

7. Oct 17, 2012

### Rably

Oh wow, I think it makes sense now. Seeing the step between x^6 and |x^3| was really helpful. I wasn't writing that step down, I was simply skipping to the final step. Thanks a bunch.