To simplify the radical expression [tex]\(\frac{1}{\sqrt{19}-\sqrt{2}}\)[/tex] by using the conjugate, follow these steps:
1. Identify the Conjugate:
The conjugate of the denominator [tex]\(\sqrt{19} - \sqrt{2}\)[/tex] is [tex]\(\sqrt{19} + \sqrt{2}\)[/tex].
2. Multiply by the Conjugate:
Multiply both the numerator and the denominator of the expression by this conjugate to rationalize the denominator:
[tex]\[
\frac{1}{\sqrt{19}-\sqrt{2}} \times \frac{\sqrt{19} + \sqrt{2}}{\sqrt{19} + \sqrt{2}}
\][/tex]
3. Simplify the Numerator:
The numerator becomes:
[tex]\[
1 \times (\sqrt{19} + \sqrt{2}) = \sqrt{19} + \sqrt{2}
\][/tex]
4. Simplify the Denominator:
The denominator is a product of two conjugates:
[tex]\[
(\sqrt{19} - \sqrt{2})(\sqrt{19} + \sqrt{2})
\][/tex]
According to the difference of squares formula [tex]\((a - b)(a + b) = a^2 - b^2\)[/tex], where [tex]\(a = \sqrt{19}\)[/tex] and [tex]\(b = \sqrt{2}\)[/tex], the expression simplifies as follows:
[tex]\[
(\sqrt{19})^2 - (\sqrt{2})^2 = 19 - 2 = 17
\][/tex]
5. Combine the Results:
Thus, the rationalized form of the given expression is:
[tex]\[
\frac{\sqrt{19} + \sqrt{2}}{17}
\][/tex]
Therefore, the simplified and rationalized form of [tex]\(\frac{1}{\sqrt{19} - \sqrt{2}}\)[/tex] is:
[tex]\[
\frac{\sqrt{19} + \sqrt{2}}{17}
\][/tex]
