Question

question
assuming x and y are both positive, write the following expression in simplest radical form.
$5y^2\sqrt{75x^2y^2}$

Explanation:

Step1: Factor the radicand

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

Step2: Simplify the square root

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a,b\geq0\)) and \(\sqrt{a^{2}} = a\) (for \(a\geq0\)), we have \(\sqrt{75x^{2}y^{2}}=\sqrt{25\times3\times x^{2}\times y^{2}}=\sqrt{25}\times\sqrt{3}\times\sqrt{x^{2}}\times\sqrt{y^{2}} = 5xy\sqrt{3}\) (since \(x\) and \(y\) are positive).

Step3: Multiply with the coefficient outside

Now, multiply \(5y^{2}\) with \(5xy\sqrt{3}\). So, \(5y^{2}\times5xy\sqrt{3}=(5\times5)xy\times y^{2}\sqrt{3}=25xy^{3}\sqrt{3}\).

Answer:

\(25xy^{3}\sqrt{3}\)