Question

11
rationalize the radicals below. ensure your final answer is as simplified as possible:
\\(\frac{2\sqrt{5}}{\sqrt{3}} =\\)
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Explanation:

Step1: Identify the radical to rationalize

We have the fraction \(\frac{2\sqrt{5}}{\sqrt{3}}\). To rationalize the denominator, we need to eliminate the square root from the denominator.

Step2: Multiply numerator and denominator by \(\sqrt{3}\)

To rationalize the denominator, we multiply the numerator and the denominator by \(\sqrt{3}\) (since \(\sqrt{a}\times\sqrt{a} = a\) for \(a\geq0\)). So we get:

$$ \frac{2\sqrt{5}\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}} $$

Step3: Simplify the numerator and the denominator

First, simplify the numerator: \(\sqrt{5}\times\sqrt{3}=\sqrt{5\times3}=\sqrt{15}\), so the numerator becomes \(2\sqrt{15}\).
Then, simplify the denominator: \(\sqrt{3}\times\sqrt{3}=\sqrt{3\times3}=\sqrt{9} = 3\).
So the fraction simplifies to \(\frac{2\sqrt{15}}{3}\).

Answer:

\(\frac{2\sqrt{15}}{3}\)