Question

simplify (sqrt{a^3b^2}) (the image shows the square root of (a^3b^2) with a question mark below)

Explanation:

Step1: Rewrite the radicand

We can rewrite \(a^{3}b^{2}\) as \(a^{2}\cdot a\cdot b^{2}\), so \(\sqrt{a^{3}b^{2}}=\sqrt{a^{2}\cdot a\cdot b^{2}}\).

Step2: Apply the square - root property

According to the property \(\sqrt{xy}=\sqrt{x}\cdot\sqrt{y}\) (\(x\geq0,y\geq0\)) and \(\sqrt{x^{2}} = |x|\) (for real numbers \(x\)), we have:
\(\sqrt{a^{2}\cdot a\cdot b^{2}}=\sqrt{a^{2}}\cdot\sqrt{b^{2}}\cdot\sqrt{a}\)
Since \(\sqrt{a^{2}}=|a|\) and \(\sqrt{b^{2}} = |b|\), and assuming \(a\geq0\) (if we are dealing with real - valued non - negative square roots in the context of simplifying algebraic expressions, we can drop the absolute value for \(a\) and \(b\) if we assume \(a\geq0\) and \(b\) is a real number. Usually, when simplifying such expressions, we assume the variables are non - negative in the domain of the square root), then \(\sqrt{a^{2}}=a\) and \(\sqrt{b^{2}}=|b|\) (or \(b\) if \(b\geq0\)). So \(\sqrt{a^{2}}\cdot\sqrt{b^{2}}\cdot\sqrt{a}=a\cdot|b|\cdot\sqrt{a}\) or \(ab\sqrt{a}\) (if \(b\geq0\)).

Answer:

If we assume \(a\geq0\) and \(b\geq0\), the simplified form is \(ab\sqrt{a}\)