Sure, let's break down the problem step-by-step to find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] for the given fraction [tex]\(\frac{4+3 \sqrt{5}}{4-3 \sqrt{5}}\)[/tex].
We need to rationalize the denominator of the fraction [tex]\(\frac{4+3 \sqrt{5}}{4-3 \sqrt{5}}\)[/tex]. To do this, we'll multiply both the numerator and the denominator by the conjugate of the denominator, which is [tex]\(4 + 3 \sqrt{5}\)[/tex].
So we have:
[tex]\[
\frac{4+3 \sqrt{5}}{4-3 \sqrt{5}} \times \frac{4+3 \sqrt{5}}{4+3 \sqrt{5}} = \frac{(4+3 \sqrt{5})^2}{(4-3 \sqrt{5})(4+3 \sqrt{5})}
\][/tex]
Let's first simplify the denominator:
[tex]\[
(4-3\sqrt{5})(4+3\sqrt{5}) = 4^2 - (3\sqrt{5})^2 = 16 - 45 = -29
\][/tex]
Next, we simplify the numerator:
[tex]\[
(4+3\sqrt{5})^2 = 4^2 + 2 \cdot 4 \cdot 3\sqrt{5} + (3\sqrt{5})^2 = 16 + 24\sqrt{5} + 45 = 61 + 24\sqrt{5}
\][/tex]
Now we have the fraction:
[tex]\[
\frac{61 + 24\sqrt{5}}{-29}
\][/tex]
This simplifies to:
[tex]\[
\frac{61}{-29} + \frac{24\sqrt{5}}{-29} = -\frac{61}{29} - \frac{24\sqrt{5}}{29}
\][/tex]
Therefore, we can identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] by comparing this expression to [tex]\(a + b\sqrt{5}\)[/tex]:
[tex]\[
a = -\frac{61}{29}, \quad b = -\frac{24}{29}
\][/tex]
Calculating these values, we get:
[tex]\[
a \approx -2.103448275862069, \quad b \approx -0.8275862068965517
\][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = -2.103448275862069, \quad b = -0.8275862068965517
\][/tex]
