Square Roots: Variables with Exponents.

Question

Alright, so let me get this straight:

$\sqrt{x^2} = |x|$

$\sqrt{x^3} = x\sqrt{x}$

$\sqrt{x^4} = x^2$

$\sqrt{x^6} = |x^3|$

Are these correct?

Answer 1

Well The first and last one are more tricky than the other two.

you know that $x^2$ is always positive and so when we take the square root it will be also positive, This is why $$\sqrt{x^2} = |x|$$

And the last one follows from the first one, But here you have to know that the an even power is always positive $x^{2n}$ is always positive for any positive integer $n$ and since $6$ is even then $x^6$ is positive and so $$\sqrt{x^6} = |(x^6)^\frac{1}{2}| = |x^3|$$

Notice that we evaluate inside out, so we first take $x^6$ then we apply the square root that's why we get the absolute value equivalence.

The 2nd and 3rd ones are very easy.

$$\sqrt{x^3} = (x^3)^\frac{1}{2} = x^\frac{3}{2} = xx^\frac{1}{2} = x\sqrt{x}$$

and the last one follows the same way

You are right !