Question

rewrite the radical expression in its simplified form,
\\(\sqrt{50x^4y^3} = \\) type your answer... \\(\sqrt{\\) type your answer...}

Explanation:

Step1: Factor the radicand

Factor \(50x^{4}y^{3}\) into perfect squares and remaining factors. We know that \(50 = 25\times2\), \(x^{4}=(x^{2})^{2}\), and \(y^{3}=y^{2}\times y\). So, \(50x^{4}y^{3}=25\times2\times(x^{2})^{2}\times y^{2}\times y\)

Step2: Apply the square - root property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (\(a\geq0,b\geq0\))

\(\sqrt{50x^{4}y^{3}}=\sqrt{25\times(x^{2})^{2}\times y^{2}\times2y}\)
Using the property \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\), we can split the square - root: \(\sqrt{25}\times\sqrt{(x^{2})^{2}}\times\sqrt{y^{2}}\times\sqrt{2y}\)

Step3: Simplify each square - root

We know that \(\sqrt{25} = 5\), \(\sqrt{(x^{2})^{2}}=x^{2}\), \(\sqrt{y^{2}} = |y|\). Since we are probably dealing with real - valued expressions and if we assume \(y\geq0\) (for simplicity, as the problem is about simplifying the radical), \(\sqrt{y^{2}}=y\).
So, \(\sqrt{25}\times\sqrt{(x^{2})^{2}}\times\sqrt{y^{2}}\times\sqrt{2y}=5\times x^{2}\times y\times\sqrt{2y}=5x^{2}y\sqrt{2y}\)

Answer:

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