Question

1. \\(\frac{4\sqrt{2}}{3\sqrt{5}}\\)

Explanation:

Step1: Rationalize the denominator

To rationalize the denominator of \(\frac{4\sqrt{2}}{3\sqrt{5}}\), we multiply the numerator and denominator by \(\sqrt{5}\) (the conjugate of the denominator's radical part).
\[
\frac{4\sqrt{2} \times \sqrt{5}}{3\sqrt{5} \times \sqrt{5}}
\]

Step2: Simplify the radicals

Using the property \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\) for the numerator and \(\sqrt{a}\times\sqrt{a} = a\) for the denominator:

• Numerator: \(4\sqrt{2\times5}=4\sqrt{10}\)
• Denominator: \(3\times5 = 15\)

So the expression becomes \(\frac{4\sqrt{10}}{15}\)

Answer:

\(\frac{4\sqrt{10}}{15}\)