Question

simplify the following expression.
\sqrt{x^{5}y^{5}} \cdot 3\sqrt{2x^{9}y^{4}}

\sqrt{x^{5}y^{5}} \cdot 3\sqrt{2x^{9}y^{4}} = \square
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

Step1: Rewrite radicals as exponents

First, recall that $\sqrt[n]{a^m}=a^{\frac{m}{n}}$. So we can rewrite the square roots (which are $n = 2$) as exponents.
The first term $\sqrt{x^{5}y^{5}}=(x^{5}y^{5})^{\frac{1}{2}}=x^{\frac{5}{2}}y^{\frac{5}{2}}$
The second term $3\sqrt{2x^{9}y^{4}} = 3\times(2x^{9}y^{4})^{\frac{1}{2}}=3\times2^{\frac{1}{2}}x^{\frac{9}{2}}y^{2}$

Step2: Multiply the two terms

When multiplying terms with the same base, we add the exponents. So multiply the coefficients and the variables separately.
First, multiply the coefficients: $1\times3\times2^{\frac{1}{2}} = 3\sqrt{2}$ (since $2^{\frac{1}{2}}=\sqrt{2}$)
For the $x$ terms: $x^{\frac{5}{2}}\times x^{\frac{9}{2}}=x^{\frac{5 + 9}{2}}=x^{\frac{14}{2}}=x^{7}$
For the $y$ terms: $y^{\frac{5}{2}}\times y^{2}=y^{\frac{5}{2}+\frac{4}{2}}=y^{\frac{9}{2}}$

Step3: Rewrite $y^{\frac{9}{2}}$ as a radical

Recall that $a^{\frac{m}{n}}=\sqrt[n]{a^m}$. So $y^{\frac{9}{2}}=\sqrt{y^{9}}$. But we can simplify $\sqrt{y^{9}}$ further. Since $y^{9}=y^{8}\times y=(y^{4})^{2}\times y$, then $\sqrt{y^{9}} = y^{4}\sqrt{y}$

Step4: Combine all the terms

Now we combine the coefficient, the $x$ term, and the simplified $y$ term.
We have $3\sqrt{2}\times x^{7}\times y^{4}\sqrt{y}$. But we can also multiply the radicals: $\sqrt{2}\times\sqrt{y}=\sqrt{2y}$
Wait, actually, let's go back to step 2. Maybe a better way is to multiply the two square roots first before dealing with exponents.

Alternative Step 2: Multiply the two square roots first
$\sqrt{x^{5}y^{5}}\times\sqrt{2x^{9}y^{4}}=\sqrt{(x^{5}y^{5})\times(2x^{9}y^{4})}=\sqrt{2x^{5 + 9}y^{5+4}}=\sqrt{2x^{14}y^{9}}$
Then multiply by 3: $3\sqrt{2x^{14}y^{9}}$

Step5: Simplify the square root

Simplify $\sqrt{2x^{14}y^{9}}$. We can break it down:
$x^{14}=(x^{7})^{2}$, so $\sqrt{x^{14}} = x^{7}$
$y^{9}=y^{8}\times y=(y^{4})^{2}\times y$, so $\sqrt{y^{9}}=y^{4}\sqrt{y}$
So $\sqrt{2x^{14}y^{9}}=x^{7}y^{4}\sqrt{2y}$
Then multiply by 3: $3x^{7}y^{4}\sqrt{2y}$

Answer:

$3x^{7}y^{4}\sqrt{2y}$