Question

express in simplest form with a rational denominator.
\\(\dfrac{5}{\sqrt{90}}\\)

Explanation:

Step1: Simplify the square root in the denominator

First, we factorize \( 90 \) to simplify \( \sqrt{90} \). We know that \( 90 = 9\times10 \), and \( \sqrt{9\times10}=\sqrt{9}\times\sqrt{10} = 3\sqrt{10} \). So the expression becomes \( \frac{5}{3\sqrt{10}} \).

Step2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by \( \sqrt{10} \) (the radical in the denominator).

$$ \frac{5\times\sqrt{10}}{3\sqrt{10}\times\sqrt{10}} $$

Simplify the denominator: \( \sqrt{10}\times\sqrt{10}=10 \), so the expression is \( \frac{5\sqrt{10}}{3\times10} \).

Step3: Simplify the fraction

Simplify \( \frac{5\sqrt{10}}{30} \) by dividing the numerator and the denominator by \( 5 \). We get \( \frac{\sqrt{10}}{6} \).

Answer:

\(\dfrac{\sqrt{10}}{6}\)