To solve the equation [tex]\(\frac{3+\sqrt{7}}{3-4\sqrt{7}} = a + b\sqrt{7}\)[/tex] where [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are rational numbers, we will need to rationalize the denominator. Here’s the step-by-step process:
1. Identify the conjugate of the denominator: The conjugate of [tex]\(3 - 4\sqrt{7}\)[/tex] is [tex]\(3 + 4\sqrt{7}\)[/tex].
2. Multiply the numerator and the denominator by this conjugate to rationalize the denominator:
[tex]\[
\frac{(3+\sqrt{7})(3+4\sqrt{7})}{(3-4\sqrt{7})(3+4\sqrt{7})}
\][/tex]
3. Expand the numerator by applying the distributive property:
[tex]\[
(3 + \sqrt{7})(3 + 4\sqrt{7}) = 3(3) + 3(4\sqrt{7}) + \sqrt{7}(3) + \sqrt{7}(4\sqrt{7})
\][/tex]
[tex]\[
= 9 + 12\sqrt{7} + 3\sqrt{7} + 4 \cdot 7
\][/tex]
[tex]\[
= 9 + 15\sqrt{7} + 28
\][/tex]
[tex]\[
= 37 + 15\sqrt{7}
\][/tex]
4. Expand the denominator:
[tex]\[
(3 - 4\sqrt{7})(3 + 4\sqrt{7}) = 3(3) - 4\sqrt{7}(4\sqrt{7})
\][/tex]
[tex]\[
= 9 - 16(7)
\][/tex]
[tex]\[
= 9 - 112
\][/tex]
[tex]\[
= -103
\][/tex]
5. Write the rationalized fraction:
[tex]\[
\frac{37 + 15\sqrt{7}}{-103}
\][/tex]
6. Separate the rational and irrational parts by splitting the fraction:
[tex]\[
\frac{37}{-103} + \frac{15\sqrt{7}}{-103}
\][/tex]
7. Simplify each term:
[tex]\[
\frac{37}{-103} = -\frac{37}{103}, \quad \frac{15\sqrt{7}}{-103} = -\frac{15\sqrt{7}}{103}
\][/tex]
8. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
By comparing the simplified fraction to [tex]\(a + b\sqrt{7}\)[/tex], we get:
[tex]\[
a = -\frac{37}{103}, \quad b = -\frac{15}{103}
\][/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[
a = -\frac{37}{103}, \quad b = -\frac{15}{103}
\][/tex]
