Answer :
To rationalize the denominator of the fraction [tex]\(\frac{5}{3 + \sqrt{6}}\)[/tex], we'll follow these steps:
1. Identify the conjugate of the denominator. The conjugate of [tex]\(3 + \sqrt{6}\)[/tex] is [tex]\(3 - \sqrt{6}\)[/tex].
2. Multiply both the numerator and the denominator of the fraction by this conjugate to eliminate the square root in the denominator.
[tex]\[ \frac{5}{3 + \sqrt{6}} \times \frac{3 - \sqrt{6}}{3 - \sqrt{6}} \][/tex]
3. Simplify the numerator:
[tex]\[ 5 \times (3 - \sqrt{6}) = 5 \cdot 3 - 5 \cdot \sqrt{6} = 15 - 5\sqrt{6} \][/tex]
4. Simplify the denominator by multiplying:
[tex]\[ (3 + \sqrt{6})(3 - \sqrt{6}) = 3^2 - (\sqrt{6})^2 = 9 - 6 = 3 \][/tex]
Thus, the rationalized fraction is:
[tex]\[ \frac{15 - 5\sqrt{6}}{3} \][/tex]
And the new denominator of the fraction is [tex]\(3\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{3}\)[/tex].
1. Identify the conjugate of the denominator. The conjugate of [tex]\(3 + \sqrt{6}\)[/tex] is [tex]\(3 - \sqrt{6}\)[/tex].
2. Multiply both the numerator and the denominator of the fraction by this conjugate to eliminate the square root in the denominator.
[tex]\[ \frac{5}{3 + \sqrt{6}} \times \frac{3 - \sqrt{6}}{3 - \sqrt{6}} \][/tex]
3. Simplify the numerator:
[tex]\[ 5 \times (3 - \sqrt{6}) = 5 \cdot 3 - 5 \cdot \sqrt{6} = 15 - 5\sqrt{6} \][/tex]
4. Simplify the denominator by multiplying:
[tex]\[ (3 + \sqrt{6})(3 - \sqrt{6}) = 3^2 - (\sqrt{6})^2 = 9 - 6 = 3 \][/tex]
Thus, the rationalized fraction is:
[tex]\[ \frac{15 - 5\sqrt{6}}{3} \][/tex]
And the new denominator of the fraction is [tex]\(3\)[/tex].
Therefore, the correct answer is [tex]\(\boxed{3}\)[/tex].
