Question

use the quotient rule to simplify. assume that all variables represent positive real numbers.
\sqrt{\frac{x^6 y}{144}}
\sqrt{\frac{x^6 y}{144}} = \square
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact answer, using

Explanation:

Step1: Apply Quotient Rule for Radicals

The quotient rule for radicals states that $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ (where $a\geq0, b > 0$). So, we can rewrite $\sqrt{\frac{x^{6}y}{144}}$ as $\frac{\sqrt{x^{6}y}}{\sqrt{144}}$.

Step2: Simplify the Denominator

We know that $\sqrt{144} = 12$ since $12\times12 = 144$.

Step3: Simplify the Numerator

For the numerator $\sqrt{x^{6}y}$, we can use the property $\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}$ (where $a\geq0, b\geq0$). So, $\sqrt{x^{6}y}=\sqrt{x^{6}}\cdot\sqrt{y}$. And $\sqrt{x^{6}}=x^{3}$ because $(x^{3})^{2}=x^{6}$. So the numerator simplifies to $x^{3}\sqrt{y}$.

Step4: Combine Numerator and Denominator

Putting it all together, we have $\frac{x^{3}\sqrt{y}}{12}$.

Answer:

$\frac{x^{3}\sqrt{y}}{12}$