To rationalize the denominator of the expression [tex]\(\frac{1}{\sqrt{9} - \sqrt{8}}\)[/tex], we'll follow these steps:
1. Identify the Conjugate: The conjugate of [tex]\(\sqrt{9} - \sqrt{8}\)[/tex] is [tex]\(\sqrt{9} + \sqrt{8}\)[/tex].
2. Multiply the Numerator and the Denominator by the Conjugate:
[tex]\[
\frac{1}{\sqrt{9} - \sqrt{8}} \cdot \frac{\sqrt{9} + \sqrt{8}}{\sqrt{9} + \sqrt{8}}
\][/tex]
3. Simplify the Denominator:
We use the difference of squares formula: [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex],
where [tex]\(a = \sqrt{9}\)[/tex] and [tex]\(b = \sqrt{8}\)[/tex]:
[tex]\[
(\sqrt{9} - \sqrt{8})(\sqrt{9} + \sqrt{8}) = (\sqrt{9})^2 - (\sqrt{8})^2 = 9 - 8 = 1
\][/tex]
4. Substitute and Simplify:
The expression now becomes:
[tex]\[
\frac{\sqrt{9} + \sqrt{8}}{1} = \sqrt{9} + \sqrt{8}
\][/tex]
5. Final Simplification:
[tex]\[
\sqrt{9} + \sqrt{8} = 3 + \sqrt{8}
\][/tex]
So, the rationalized form of [tex]\(\frac{1}{\sqrt{9} - \sqrt{8}}\)[/tex] is [tex]\(3 + \sqrt{8}\)[/tex].
