Question

assuming ( x ) and ( y ) are both positive, write the following expression in simplest radical form. (sqrt{75x^{3}y^{6}})

Explanation:

Step1: Factor the radicand

Factor \(75x^{3}y^{6}\) into perfect squares and other factors. We know that \(75 = 25\times3\), \(x^{3}=x^{2}\times x\), and \(y^{6}=(y^{3})^{2}\). So, \(\sqrt{75x^{3}y^{6}}=\sqrt{25\times3\times x^{2}\times x\times(y^{3})^{2}}\).

Step2: Apply the square - root property

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\)) and \(\sqrt{a^{2}} = a\) (\(a\geq0\)), we have:
\(\sqrt{25\times3\times x^{2}\times x\times(y^{3})^{2}}=\sqrt{25}\times\sqrt{x^{2}}\times\sqrt{(y^{3})^{2}}\times\sqrt{3x}\)
Since \(\sqrt{25} = 5\), \(\sqrt{x^{2}}=x\) (because \(x>0\)) and \(\sqrt{(y^{3})^{2}}=y^{3}\) (because \(y > 0\)), the expression becomes \(5\times x\times y^{3}\times\sqrt{3x}\).

Step3: Simplify the expression

Simplify \(5\times x\times y^{3}\times\sqrt{3x}\) to \(5xy^{3}\sqrt{3x}\).

Answer:

\(5xy^{3}\sqrt{3x}\)