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Rationalize the denominator of the fraction below. What is the new denominator?

[tex]\[\frac{5}{3+\sqrt{6}}\][/tex]

A. 3
B. 15
C. -27
D. -3

Sagot :

GinnyAnswer
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].
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