Question

expressions 40 72% \sqrt{a^{3}b^{2}} ?

Explanation:

Step1: Simplify the square root

We can use the property of square roots $\sqrt{xy}=\sqrt{x}\cdot\sqrt{y}$ (where $x\geq0,y\geq0$) and $\sqrt{x^n}=x^{\frac{n}{2}}$ for non - negative $x$.
First, rewrite $a^{3}$ as $a^{2}\cdot a$ and $b^{2}$ remains as it is. So $\sqrt{a^{3}b^{2}}=\sqrt{a^{2}\cdot a\cdot b^{2}}$.

Step2: Apply the square root property

Using $\sqrt{x\cdot y}=\sqrt{x}\cdot\sqrt{y}$, we get $\sqrt{a^{2}\cdot a\cdot b^{2}}=\sqrt{a^{2}}\cdot\sqrt{a}\cdot\sqrt{b^{2}}$.
Since $\sqrt{x^{2}} = |x|$, for $a\geq0$ and $b$ (we assume $b$ is a real number, and if we consider the principal square root, for $b^{2}$, $\sqrt{b^{2}}=|b|$, but if we assume $b\geq0$, then $\sqrt{b^{2}} = b$ and $\sqrt{a^{2}}=a$ (when $a\geq0$)). So $\sqrt{a^{2}}\cdot\sqrt{a}\cdot\sqrt{b^{2}}=a\cdot\sqrt{a}\cdot|b|$. If we assume $b\geq0$, then it simplifies to $ab\sqrt{a}$.

Answer:

If we assume \(a\geq0\) and \(b\geq0\), the simplified form is \(ab\sqrt{a}\) (or \(|b|a\sqrt{a}\) if we consider the absolute value for \(b\)).