Simplify radical expressions with variable (absolute value)

Question

I am having trouble understanding how in the cases I listed that certain terms are taking the absolute value.

√25x = 5 |x^3|

√900x^2 = 30 |x|

√729x^3 = 27 |x| √x

√400x^14 = 4 |x|

I can easily do all other examples that do not rely on taking the absolute value of certain terms. What am I not seeing? Please explain in simplest way, if possible.

Thank You, Brian

Answer 1

When taking an even root (i.e. a square root), your result has to be positive.

We might think that √(900x²) = 30x However, we'll run into a problem if we plug in a negative number.

For example, plug in -1 for x:

On the left side of the equation, you get √( 900(-1)²) = √900 = 30

On the right side of the equation, you get 30(-1) = -30.

The left side does not equal the right, but it can be changed if we just make the right side positive.

And that's why you use absolute values around the x