Question

simplify (sqrt{xyz^2})

Explanation:

Step1: Recall square - root property

The square - root property states that \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (for \(a\geq0,b\geq0\)) and \(\sqrt{x^{2}} = |x|\) (in general) or \(x\) if \(x\geq0\). Also, \(\sqrt{x^{n}}=x^{\frac{n}{2}}\) when \(x\geq0\). For the expression \(\sqrt{xyz^{2}}\), we can use the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) to separate the factors inside the square root.
We know that \(\sqrt{xyz^{2}}=\sqrt{x}\cdot\sqrt{y}\cdot\sqrt{z^{2}}\) (assuming \(x\geq0,y\geq0,z\in R\)).

Step2: Simplify \(\sqrt{z^{2}}\)

According to the property of square - roots, \(\sqrt{z^{2}} = |z|\). But if we assume that \(z\) is non - negative (or we are working in a context where we can take the principal square root and \(z\geq0\)), then \(\sqrt{z^{2}}=z\).
So, \(\sqrt{xyz^{2}}=\sqrt{xy}\cdot z=z\sqrt{xy}\) (when \(z\geq0,x\geq0,y\geq0\)). If we consider the general case with absolute value, it is \(|z|\sqrt{xy}\). But usually, in basic algebra problems, we assume the variables are non - negative for the principal square root, so the simplified form is \(z\sqrt{xy}\).

Answer:

If we assume \(x\geq0,y\geq0,z\geq0\), the simplified form of \(\sqrt{xyz^{2}}\) is \(z\sqrt{xy}\). In the general case (with absolute value), it is \(|z|\sqrt{xy}\).