Question

assuming x and y are both positive, write the following expression in simplest radical form.
\\( y^2 \sqrt{25x^7 y^7} \\)

Explanation:

Step1: Simplify the square root

First, we simplify the square root \(\sqrt{25x^{7}y^{7}}\). We know that \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (for \(a,b\geq0\)) and \(\sqrt{a^{2}} = a\) (for \(a\geq0\)).

For the coefficient: \(\sqrt{25}=5\).

For the \(x\)-term: \(x^{7}=x^{6}\cdot x=(x^{3})^{2}\cdot x\), so \(\sqrt{x^{7}}=\sqrt{(x^{3})^{2}\cdot x}=x^{3}\sqrt{x}\) (since \(x\) is positive).

For the \(y\)-term: \(y^{7}=y^{6}\cdot y=(y^{3})^{2}\cdot y\), so \(\sqrt{y^{7}}=\sqrt{(y^{3})^{2}\cdot y}=y^{3}\sqrt{y}\) (since \(y\) is positive).

Putting it all together, \(\sqrt{25x^{7}y^{7}} = 5x^{3}y^{3}\sqrt{xy}\).

Step2: Multiply by \(y^{2}\)

Now we multiply this result by \(y^{2}\): \(y^{2}\cdot5x^{3}y^{3}\sqrt{xy}\). Using the rule of exponents \(a^{m}\cdot a^{n}=a^{m + n}\) for \(y\)-terms: \(y^{2}\cdot y^{3}=y^{2 + 3}=y^{5}\).

So the expression becomes \(5x^{3}y^{5}\sqrt{xy}\).

Answer:

\(5x^{3}y^{5}\sqrt{xy}\)