Question

given all variables below are positive, express in simplest radical form with a rational denominator.\\(\frac{\sqrt{16x^2y^{11}}}{\sqrt{x^2y}}\\)

Explanation:

Step1: Combine the radicals

Using the property $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$, we combine the two square roots:
$$\frac{\sqrt{16x^{2}y^{11}}}{\sqrt{x^{2}y}} = \sqrt{\frac{16x^{2}y^{11}}{x^{2}y}}$$

Step2: Simplify the fraction inside the radical

Simplify the variables and coefficients:

• The $x^{2}$ terms cancel out: $\frac{x^{2}}{x^{2}} = 1$
• For the $y$ terms: $\frac{y^{11}}{y} = y^{10}$ (using the rule $\frac{a^{m}}{a^{n}} = a^{m - n}$, so $11 - 1 = 10$)
• The coefficient is $\frac{16}{1} = 16$

So we have $\sqrt{16y^{10}}$

Step3: Simplify the square root

• $\sqrt{16} = 4$
• $\sqrt{y^{10}} = y^{5}$ (since $\sqrt{a^{2n}} = a^{n}$, here $2n = 10$ so $n = 5$)

Multiplying these together gives $4y^{5}$

Answer:

$4y^{5}$