Question

assuming $x$ and $y$ are both positive, write the following expression in simplest radical form.

$5y\sqrt{25x^5y^2}$

Explanation:

Step1: Simplify the square root part

First, we can break down the radicand \(25x^{5}y^{2}\) into perfect squares and remaining factors. We know that \(25 = 5^{2}\), \(x^{5}=x^{4}\cdot x=(x^{2})^{2}\cdot x\), and \(y^{2}=(y)^{2}\). So, \(\sqrt{25x^{5}y^{2}}=\sqrt{5^{2}\cdot(x^{2})^{2}\cdot x\cdot y^{2}}\).
Using the property of square roots \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) (for \(a,b\geq0\)) and \(\sqrt{a^{2}} = a\) (for \(a\geq0\)), we get:
\(\sqrt{5^{2}\cdot(x^{2})^{2}\cdot x\cdot y^{2}}=\sqrt{5^{2}}\cdot\sqrt{(x^{2})^{2}}\cdot\sqrt{y^{2}}\cdot\sqrt{x}=5\cdot x^{2}\cdot y\cdot\sqrt{x}\) (since \(x\) and \(y\) are positive, we can take the positive square roots).

Step2: Multiply with the outside factor

Now, we have the original expression \(5y\sqrt{25x^{5}y^{2}}\). Substituting the simplified square root from Step 1, we get:
\(5y\cdot(5x^{2}y\sqrt{x})\)
Now, multiply the coefficients and the like - variable terms. The coefficients are \(5\) and \(5\), so \(5\times5 = 25\). For the \(y\) terms, \(y\times y=y^{2}\). And we still have the \(x^{2}\) and \(\sqrt{x}\) terms. So, \(5y\cdot(5x^{2}y\sqrt{x})=25x^{2}y^{2}\sqrt{x}\).

Answer:

\(25x^{2}y^{2}\sqrt{x}\)