Question

express in simplest form with a rational denominator.
\\(\frac{5}{\sqrt{45}}\\)
answer attempt 1 out of 10

Explanation:

Step1: Simplify the square root in the denominator

First, we simplify \(\sqrt{45}\). We know that \(45 = 9\times5\), and \(\sqrt{9\times5}=\sqrt{9}\times\sqrt{5}=3\sqrt{5}\) (since \(\sqrt{9} = 3\)). So the expression becomes \(\frac{5}{3\sqrt{5}}\).

Step2: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{5}\) (because the denominator has \(\sqrt{5}\), and multiplying by \(\sqrt{5}\) will make the denominator a rational number).

So, \(\frac{5\times\sqrt{5}}{3\sqrt{5}\times\sqrt{5}}\).

Now, simplify the denominator: \(\sqrt{5}\times\sqrt{5}=5\) (by the property \(\sqrt{a}\times\sqrt{a}=a\) for \(a\geq0\)).

The numerator is \(5\sqrt{5}\) and the denominator is \(3\times5 = 15\). So the expression is \(\frac{5\sqrt{5}}{15}\).

Step3: Simplify the fraction

We can simplify \(\frac{5\sqrt{5}}{15}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 5.

Dividing the numerator: \(5\sqrt{5}\div5=\sqrt{5}\)

Dividing the denominator: \(15\div5 = 3\)

So the simplified form is \(\frac{\sqrt{5}}{3}\).

Answer:

\(\frac{\sqrt{5}}{3}\)