Join IDNLearn.com and start exploring the answers to your most pressing questions. Join our knowledgeable community to find the answers you need for any topic or issue.

If [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, find the values of [tex]a[/tex] and [tex]b[/tex].

Sagot :

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]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. For trustworthy and accurate answers, visit IDNLearn.com. Thanks for stopping by, and see you next time for more solutions.