Answer :
To rationalize the denominator of the given expression [tex]\(\frac{\sqrt{5}}{3 \sqrt{3}}\)[/tex], follow these steps:
1. Identify the denominator: The denominator is [tex]\(3 \sqrt{3}\)[/tex].
2. Multiply the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]: This is done to eliminate the square root in the denominator.
[tex]\[ \frac{\sqrt{5}}{3 \sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \][/tex]
3. Perform the multiplication: Multiply both the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[ \frac{\sqrt{5} \cdot \sqrt{3}}{3 \sqrt{3} \cdot \sqrt{3}} \][/tex]
4. Simplify the expressions:
- The numerator becomes [tex]\(\sqrt{5} \cdot \sqrt{3} = \sqrt{5 \cdot 3} = \sqrt{15}\)[/tex].
- The denominator becomes [tex]\(3 \sqrt{3} \cdot \sqrt{3} = 3 \cdot (\sqrt{3})^2 = 3 \cdot 3 = 9\)[/tex].
Thus, the fraction simplifies to:
[tex]\[ \frac{\sqrt{15}}{9} \][/tex]
Therefore, the expression with a rationalized denominator is:
[tex]\[ \frac{\sqrt{15}}{9} \][/tex]
1. Identify the denominator: The denominator is [tex]\(3 \sqrt{3}\)[/tex].
2. Multiply the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]: This is done to eliminate the square root in the denominator.
[tex]\[ \frac{\sqrt{5}}{3 \sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} \][/tex]
3. Perform the multiplication: Multiply both the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[ \frac{\sqrt{5} \cdot \sqrt{3}}{3 \sqrt{3} \cdot \sqrt{3}} \][/tex]
4. Simplify the expressions:
- The numerator becomes [tex]\(\sqrt{5} \cdot \sqrt{3} = \sqrt{5 \cdot 3} = \sqrt{15}\)[/tex].
- The denominator becomes [tex]\(3 \sqrt{3} \cdot \sqrt{3} = 3 \cdot (\sqrt{3})^2 = 3 \cdot 3 = 9\)[/tex].
Thus, the fraction simplifies to:
[tex]\[ \frac{\sqrt{15}}{9} \][/tex]
Therefore, the expression with a rationalized denominator is:
[tex]\[ \frac{\sqrt{15}}{9} \][/tex]
