Question

simplify
\sqrt{x^2y^7}
options:
$xy^3$
$xy^3\sqrt{xy}$
$xy^3\sqrt{y}$

Explanation:

Step1: Rewrite the radicand

We can rewrite \(y^{7}\) as \(y^{6}\cdot y\) and \(x^{2}\) remains as it is. So, \(\sqrt{x^{2}y^{7}}=\sqrt{x^{2}\cdot y^{6}\cdot y}\).

Step2: Use the property of square roots

The property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (for \(a\geq0,b\geq0\)) allows us to split the square root: \(\sqrt{x^{2}\cdot y^{6}\cdot y}=\sqrt{x^{2}}\cdot\sqrt{y^{6}}\cdot\sqrt{y}\).

Step3: Simplify each square root

We know that \(\sqrt{x^{2}} = |x|\), but if we assume \(x\) and \(y\) are non - negative (since we are dealing with real - valued simplification in the context of a basic algebra problem), \(\sqrt{x^{2}}=x\), \(\sqrt{y^{6}} = y^{3}\) (because \((y^{3})^{2}=y^{6}\)). So, \(\sqrt{x^{2}}\cdot\sqrt{y^{6}}\cdot\sqrt{y}=x\cdot y^{3}\cdot\sqrt{y}=xy^{3}\sqrt{y}\).

Answer:

\(xy^{3}\sqrt{y}\) (the option with the orange background)