Question

divide and simplify. assume that all variables are positive.\\(\frac{\sqrt{96x^{9}y^{8}}}{\sqrt{8x^{8}y^{5}}}\\)\\(\frac{\sqrt{96x^{9}y^{8}}}{\sqrt{8x^{8}y^{5}}}=\square\\)\\((\text{type an exact answer, using radicals as needed. simplify your answer.})\\)

Explanation:

Step1: Use quotient rule for radicals

The quotient rule for radicals states that $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ (where $a\geq0$ and $b > 0$). So we can combine the two square roots into one:
$\frac{\sqrt{96x^{9}y^{8}}}{\sqrt{8x^{8}y^{5}}}=\sqrt{\frac{96x^{9}y^{8}}{8x^{8}y^{5}}}$

Step2: Simplify the fraction inside the square root

First, simplify the coefficients: $\frac{96}{8} = 12$.
Then, use the quotient rule for exponents $a^{m}\div a^{n}=a^{m - n}$ for $x$ and $y$ terms:
For $x$: $x^{9}\div x^{8}=x^{9 - 8}=x^{1}=x$
For $y$: $y^{8}\div y^{5}=y^{8 - 5}=y^{3}$
So the fraction inside the square root becomes $\frac{96x^{9}y^{8}}{8x^{8}y^{5}}=12xy^{3}$
Now our expression is $\sqrt{12xy^{3}}$

Step3: Simplify the square root

We can factor $12xy^{3}$ into perfect square factors and non - perfect square factors.
$12 = 4\times3$, and $y^{3}=y^{2}\times y$
So $\sqrt{12xy^{3}}=\sqrt{4y^{2}\times3xy}$
Using the product rule for radicals $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ (where $a\geq0$ and $b\geq0$), we get:
$\sqrt{4y^{2}}\times\sqrt{3xy}$
Since $\sqrt{4y^{2}} = 2y$ (because $y>0$), the expression simplifies to $2y\sqrt{3xy}$

Answer:

$2y\sqrt{3xy}$