Question

question
assuming x and y are both positive, write the following expression in simplest radical form.
$3x^3y\sqrt{175x^5y^2}$

Explanation:

Step1: Factor the radicand

Factor \(175x^{5}y^{2}\) into perfect squares and remaining factors. \(175 = 25\times7\), \(x^{5}=x^{4}\times x\), \(y^{2}=y^{2}\). So \(175x^{5}y^{2}=25\times7\times x^{4}\times x\times y^{2}\).

Step2: Simplify the square root

Use the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (\(a,b\geq0\)) and \(\sqrt{a^{2}} = a\) (\(a\geq0\)).
\(\sqrt{175x^{5}y^{2}}=\sqrt{25\times7\times x^{4}\times x\times y^{2}}=\sqrt{25}\cdot\sqrt{x^{4}}\cdot\sqrt{y^{2}}\cdot\sqrt{7x}=5x^{2}y\sqrt{7x}\).

Step3: Multiply with the outside term

Multiply \(3x^{3}y\) with \(5x^{2}y\sqrt{7x}\). Using the property \(a^{m}\cdot a^{n}=a^{m + n}\) for \(x\) and \(y\) terms:
\(3x^{3}y\times5x^{2}y\sqrt{7x}=(3\times5)x^{3 + 2}y^{1+1}\sqrt{7x}=15x^{5}y^{2}\sqrt{7x}\).

Answer:

\(15x^{5}y^{2}\sqrt{7x}\)