Question

use the product rule to multiply. assume that all variables represent positive real numbers.
\sqrt{\frac{2}{x}} \cdot \sqrt{\frac{5}{y}}

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

Explanation:

Step1: Apply the product rule for radicals

The product rule for radicals states that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}\) for non - negative real numbers \(a\) and \(b\). Here, \(a=\frac{2}{x}\) and \(b = \frac{5}{y}\). So we have \(\sqrt{\frac{2}{x}}\cdot\sqrt{\frac{5}{y}}=\sqrt{\frac{2}{x}\cdot\frac{5}{y}}\)

Step2: Multiply the fractions inside the radical

Multiply the numerators and denominators of the fractions \(\frac{2}{x}\) and \(\frac{5}{y}\) separately. The numerator is \(2\times5 = 10\) and the denominator is \(x\times y=xy\). So \(\sqrt{\frac{2}{x}\cdot\frac{5}{y}}=\sqrt{\frac{10}{xy}}\)

Step3: Rationalize the denominator (optional, but to present in a standard form)

We can rewrite \(\sqrt{\frac{10}{xy}}\) as \(\frac{\sqrt{10}}{\sqrt{xy}}\), and then rationalize the denominator by multiplying the numerator and denominator by \(\sqrt{xy}\). We get \(\frac{\sqrt{10}\cdot\sqrt{xy}}{\sqrt{xy}\cdot\sqrt{xy}}=\frac{\sqrt{10xy}}{xy}\) (since \(\sqrt{a}\cdot\sqrt{b}=\sqrt{ab}\) and \(\sqrt{a}\cdot\sqrt{a}=a\) for \(a\geq0\))

Answer:

\(\frac{\sqrt{10xy}}{xy}\)