Question

question
assuming x and y are both positive, write the following expression in simplest radical form.
$x^{2}y^{3}\sqrt{50x^{3}y^{5}}$

Explanation:

Step1: Simplify the square root term

First, simplify $\sqrt{50x^{3}y^{5}}$. We can factor 50 as $25\times2$, $x^{3}$ as $x^{2}\times x$, and $y^{5}$ as $y^{4}\times y$. So, $\sqrt{50x^{3}y^{5}}=\sqrt{25\times2\times x^{2}\times x\times y^{4}\times y}$. Using the property $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ (for $a,b\geq0$), we get $\sqrt{25}\times\sqrt{2}\times\sqrt{x^{2}}\times\sqrt{x}\times\sqrt{y^{4}}\times\sqrt{y}$. Since $x$ and $y$ are positive, $\sqrt{x^{2}} = x$ and $\sqrt{y^{4}}=y^{2}$. So this simplifies to $5xy^{2}\sqrt{2xy}$.

Step2: Multiply with the outside term

Now, multiply this with the outside term $x^{2}y^{3}$. So we have $x^{2}y^{3}\times5xy^{2}\sqrt{2xy}$. When multiplying variables with exponents, we add the exponents for the same base. For $x$: $x^{2}\times x = x^{2 + 1}=x^{3}$. For $y$: $y^{3}\times y^{2}=y^{3+2}=y^{5}$. And the coefficient is 5. So putting it together, we get $5x^{3}y^{5}\sqrt{2xy}$.

Answer:

$5x^{3}y^{5}\sqrt{2xy}$