Question

use the quotient rule to simplify. assume that all variables represent positive real numbers.
\sqrt{\frac{x^2y}{289}}
\sqrt{\frac{x^2y}{289}} = \square
(simplify your answer. use integers or fractions for any numbers in the expression. type an exact

Explanation:

Step1: Recall the quotient rule for square roots

The quotient rule for square roots states that \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) for \(a\geq0\) and \(b > 0\). So we can apply this rule to \(\sqrt{\frac{x^{2}y}{289}}\) and get \(\frac{\sqrt{x^{2}y}}{\sqrt{289}}\).

Step2: Simplify the square roots

We know that \(\sqrt{x^{2}y}=\sqrt{x^{2}}\cdot\sqrt{y}\) (by the product rule of square roots \(\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}\) for \(a\geq0,b\geq0\)) and since \(x\) represents a positive real number, \(\sqrt{x^{2}} = x\). Also, \(\sqrt{289}=\sqrt{17^{2}} = 17\) (because \(17\times17 = 289\)).
So substituting these values back, we have \(\frac{x\sqrt{y}}{17}\).

Answer:

\(\frac{x\sqrt{y}}{17}\)