Respuesta :

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To divide a number with a surd, we rationalize the surd.

This is done by multiplying both the numerator and the denominator of the surd with the conjugate of the surd (i.e. the surd with the sign in the middle changed).

Therefore, given

[tex] \frac{1}{1+\sqrt{3}} \\ \\ \Rightarrow \frac{1}{1+\sqrt{3}} \times \frac{1-\sqrt{3}}{1-\sqrt{3}} \\ \\ = \frac{1-\sqrt{3}}{1-3}= -\frac{1}{2} \left(1-\sqrt{3}\right) \\ \\ = \frac{1}{2} \left(\sqrt{3}-1\right)[/tex]

Answer: The following quotient of:

[tex]\frac{1}{1+\sqrt{3}}=\frac{\sqrt{3}-1}{2}[/tex]

Step-by-step explanation:

[tex]\frac{1}{1+\sqrt{3}}[/tex]

Multiply and divide by [tex]1-\sqrt{3}[/tex]

[tex]\frac{1}{1+\sqrt{3}}\times \frac{1-\sqrt{3}}{1-\sqrt{3}}[/tex]

Using identity [tex](a+b)(a-b)=a^2-b^2[/tex]

[tex]\frac{1-\sqrt{3}}{(1^2-\sqrt{3}^2)}=\frac{1-\sqrt{3}}{(1-3)}[/tex]

[tex]\frac{1-\sqrt{3}}{-2}=\frac{\sqrt{3}-1}{2}[/tex]

The following quotient of:

[tex]\frac{1}{1+\sqrt{3}}=\frac{\sqrt{3}-1}{2}[/tex]

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