Simplify the square root of (20 x^2 y^2) / z^5.

Step-by-step solution

1. 1

   Split every factor into a perfect square times a leftover.

   $20=4\cdot 5,\qquad z^5=z^4\cdot z$

   so the radicand becomes

   $\frac{4\cdot 5\cdot x^2y^2}{z^4\cdot z}$

2. 2

   Take the square roots of the perfect squares, using absolute values where needed. $\sqrt{u^2}=|u|$, not $u$:

   $\sqrt4=2,\quad\sqrt{x^2}=|x|,\quad\sqrt{y^2}=|y|,\quad\sqrt{z^4}=z^2$

   No bars are needed on $z^2$ because it is already non-negative. The result so far:

   $\frac{2|x||y|\sqrt5}{z^2\sqrt z}$

3. 3

   Note the domain. The original requires $z>0$ (an odd power of $z$ in the denominator under a square root), which is why $z$ itself needs no absolute value.

4. 4

   Rationalise the denominator. Multiply numerator and denominator by $\sqrt z$:

   $\frac{2|x||y|\sqrt5}{z^2\sqrt z}\cdot\frac{\sqrt z}{\sqrt z}=\frac{2|xy|\sqrt{5z}}{z^3}$

   since $\sqrt5\cdot\sqrt z=\sqrt{5z}$ and $z^2\cdot z=z^3$.

5. 5

   Verify numerically. Take $x=2$, $y=3$, $z=4$: the radicand is $\tfrac{20\cdot 4\cdot 9}{1024}=0.703125$, whose root is $0.83853$. The simplified form gives $\tfrac{2\cdot 6\sqrt{20}}{64}=\tfrac{12\cdot 4.47214}{64}=0.83853$ $\checkmark$. If all variables are assumed positive the answer may be written $\tfrac{2xy\sqrt{5z}}{z^3}$.