Question

simplify (sqrt{a^{3}b^{2}})

Explanation:

Step1: Break down the radicand

We can rewrite \(a^3b^2\) as \(a^2 \cdot a \cdot b^2\) since \(a^3 = a^2 \cdot a\) and \(b^2\) remains as is. So, \(\sqrt{a^3b^2}=\sqrt{a^2 \cdot a \cdot b^2}\).

Step2: Use the square - root property \(\sqrt{xy}=\sqrt{x}\cdot\sqrt{y}\) (\(x\geq0,y\geq0\))

Applying the property \(\sqrt{xy}=\sqrt{x}\cdot\sqrt{y}\), we get \(\sqrt{a^2 \cdot a \cdot b^2}=\sqrt{a^2}\cdot\sqrt{a}\cdot\sqrt{b^2}\).

Step3: Simplify each square - root term

We know that \(\sqrt{x^2}=|x|\), but if we assume \(a\geq0\) and \(b\) is a real number (usually in the context of simplifying radical expressions for basic algebra problems, we assume the variables are non - negative for simplicity), then \(\sqrt{a^2}=a\) and \(\sqrt{b^2}=|b|\) (or \(b\) if \(b\geq0\)). 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 \(a\geq0\) and \(b\geq0\), the simplified form is \(ab\sqrt{a}\)