Question

sqrt{\frac{56x^{5}y^{5}}{7xy}}

Explanation:

Step1: Simplify the fraction inside the square root

First, simplify \(\frac{56x^{5}y^{5}}{7xy}\). Divide the coefficients: \(56\div7 = 8\). For the \(x\)-terms, use the rule \(a^{m}\div a^{n}=a^{m - n}\), so \(x^{5}\div x = x^{5 - 1}=x^{4}\). For the \(y\)-terms, \(y^{5}\div y = y^{5 - 1}=y^{4}\). So the fraction simplifies to \(8x^{4}y^{4}\). Now the expression is \(\sqrt{8x^{4}y^{4}}\).

Step2: Simplify the square root

We can rewrite \(8x^{4}y^{4}\) as \(4\times2\times(x^{2})^{2}\times(y^{2})^{2}\). Then, using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a,b\geq0\)) and \(\sqrt{a^{2}} = a\) (for \(a\geq0\)), we have \(\sqrt{4\times2\times(x^{2})^{2}\times(y^{2})^{2}}=\sqrt{4}\times\sqrt{2}\times\sqrt{(x^{2})^{2}}\times\sqrt{(y^{2})^{2}}\). Simplifying each square root: \(\sqrt{4} = 2\), \(\sqrt{(x^{2})^{2}}=x^{2}\), \(\sqrt{(y^{2})^{2}}=y^{2}\), and \(\sqrt{2}\) remains as is. So putting it together, we get \(2x^{2}y^{2}\sqrt{2}\).

Answer:

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