Square Roots: Variables with Exponents.
By Sophia Hammond
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?
$\endgroup$ 41 Answer
$\begingroup$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 !
$\endgroup$ 1
