Sure! Let's solve the given problem step-by-step and find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] for the expression [tex]\(\frac{1 - \sqrt{3}}{1 + \sqrt{3}}\)[/tex].
1. Given Expression:
[tex]\[
\frac{1 - \sqrt{3}}{1 + \sqrt{3}} = a + b \sqrt{3}
\][/tex]
2. Rationalizing the Denominator:
To rationalize the denominator, we'll multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(1 + \sqrt{3}\)[/tex] is [tex]\(1 - \sqrt{3}\)[/tex].
[tex]\[
\frac{1 - \sqrt{3}}{1 + \sqrt{3}} \times \frac{1 - \sqrt{3}}{1 - \sqrt{3}} = \frac{(1 - \sqrt{3})^2}{(1 + \sqrt{3})(1 - \sqrt{3})}
\][/tex]
3. Simplify the Expression:
- Denominator:
[tex]\[
(1 + \sqrt{3})(1 - \sqrt{3}) = 1^2 - (\sqrt{3})^2 = 1 - 3 = -2
\][/tex]
- Numerator:
[tex]\[
(1 - \sqrt{3})^2 = 1^2 - 2 \cdot 1 \cdot \sqrt{3} + (\sqrt{3})^2 = 1 - 2\sqrt{3} + 3 = 4 - 2\sqrt{3}
\][/tex]
Therefore, our rationalized fraction is:
[tex]\[
\frac{4 - 2\sqrt{3}}{-2}
\][/tex]
4. Separate the Fraction:
[tex]\[
\frac{4 - 2\sqrt{3}}{-2} = \frac{4}{-2} - \frac{2\sqrt{3}}{-2} = -2 + \sqrt{3}
\][/tex]
5. Identifying [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
Comparing [tex]\(-2 + \sqrt{3}\)[/tex] with [tex]\(a + b\sqrt{3}\)[/tex], we get:
[tex]\[
a = -2
\][/tex]
[tex]\[
b = 1
\][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = -0.2679491924311228
\][/tex]
[tex]\[
b = 1.7320508075688772
\][/tex]
