Question

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

( x^2 sqrt{20x^7 y^4} )

Explanation:

Step1: Factor the radicand

First, factor the number and variables inside the square root. We know that \(20 = 4\times5\), \(x^{7}=x^{6}\times x\), and \(y^{4}=(y^{2})^{2}\). So, \(\sqrt{20x^{7}y^{4}}=\sqrt{4\times5\times x^{6}\times x\times(y^{2})^{2}}\).

Step2: Simplify the square root

Using the property \(\sqrt{ab}=\sqrt{a}\times\sqrt{b}\) (for \(a,b\geq0\)) and \(\sqrt{a^{2}} = a\) (for \(a\geq0\)), we can simplify the square root:
\[

$$\begin{align*} \sqrt{4\times5\times x^{6}\times x\times(y^{2})^{2}}&=\sqrt{4}\times\sqrt{5}\times\sqrt{x^{6}}\times\sqrt{x}\times\sqrt{(y^{2})^{2}}\\ &= 2\times\sqrt{5}\times x^{3}\times\sqrt{x}\times y^{2}\\ &= 2x^{3}y^{2}\sqrt{5x} \end{align*}$$

\]

Step3: Multiply by the outside term

Now, we multiply the outside term \(x^{2}\) with the simplified square root:
\[
x^{2}\times2x^{3}y^{2}\sqrt{5x}=2x^{2 + 3}y^{2}\sqrt{5x}=2x^{5}y^{2}\sqrt{5x}
\]

Answer:

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