Question

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

Explanation:

Assuming the problem is to rationalize the denominators of the given expressions:

For \( k. \frac{3\sqrt{5}}{\sqrt{2}} \)

Step1: Multiply numerator and denominator by \( \sqrt{2} \)

To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator (which is \( \sqrt{2} \) here since the denominator is a single square root term). So we have:
\( \frac{3\sqrt{5}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} \)

Step2: Simplify the expression

Simplify the numerator and the denominator. The numerator becomes \( 3\sqrt{5} \times \sqrt{2}=3\sqrt{10} \) (using the property \( \sqrt{a}\times\sqrt{b}=\sqrt{ab} \)) and the denominator becomes \( \sqrt{2}\times\sqrt{2} = 2 \) (using the property \( \sqrt{a}\times\sqrt{a}=a \)). So the rationalized form is \( \frac{3\sqrt{10}}{2} \)

For \( 1. \frac{1}{\sqrt{3}} \)

Step1: Multiply numerator and denominator by \( \sqrt{3} \)

To rationalize the denominator, we multiply the numerator and the denominator by \( \sqrt{3} \) (the conjugate of the denominator). So we have:
\( \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \)

Step2: Simplify the expression

Simplify the numerator and the denominator. The numerator becomes \( 1\times\sqrt{3}=\sqrt{3} \) and the denominator becomes \( \sqrt{3}\times\sqrt{3}=3 \) (using the property \( \sqrt{a}\times\sqrt{a}=a \)). So the rationalized form is \( \frac{\sqrt{3}}{3} \)

Answer:

• For \( \frac{3\sqrt{5}}{\sqrt{2}} \): \( \frac{3\sqrt{10}}{2} \)
• For \( \frac{1}{\sqrt{3}} \): \( \frac{\sqrt{3}}{3} \)