Sure, let's rationalize the denominator of the given expression [tex]\(\frac{s}{\sqrt{3} - \sqrt{5}}\)[/tex]. Here are the step-by-step instructions:
1. Identify the expression to be rationalized:
We start with the fraction [tex]\(\frac{s}{\sqrt{3} - \sqrt{5}}\)[/tex].
2. Multiply by a suitable form of 1:
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of [tex]\(\sqrt{3} - \sqrt{5}\)[/tex] is [tex]\(\sqrt{3} + \sqrt{5}\)[/tex]. So we multiply by [tex]\(\frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}}\)[/tex]:
[tex]\[
\frac{s}{\sqrt{3} - \sqrt{5}} \cdot \frac{\sqrt{3} + \sqrt{5}}{\sqrt{3} + \sqrt{5}} = \frac{s (\sqrt{3} + \sqrt{5})}{(\sqrt{3} - \sqrt{5})(\sqrt{3} + \sqrt{5})}
\][/tex]
3. Simplify the denominator:
The denominator is a difference of squares:
[tex]\[
(\sqrt{3} - \sqrt{5}) (\sqrt{3} + \sqrt{5}) = (\sqrt{3})^2 - (\sqrt{5})^2 = 3 - 5 = -2
\][/tex]
4. Combine the results:
Putting the simplified denominator back into the expression, we get:
[tex]\[
\frac{s (\sqrt{3} + \sqrt{5})}{-2}
\][/tex]
5. Simplify the fraction:
Finally, we can write the result as:
[tex]\[
\frac{s (\sqrt{3} + \sqrt{5})}{-2} = \frac{s}{-2}(\sqrt{3} + \sqrt{5})
\][/tex]
Thus, the expression [tex]\(\frac{s}{\sqrt{3} - \sqrt{5}}\)[/tex] can be rationalized to:
[tex]\[
\frac{s}{-2}(\sqrt{3} + \sqrt{5})
\][/tex]
And simplifying the negative sign:
[tex]\[
\frac{s(\sqrt{3} + \sqrt{5})}{-2} = s \left(-\frac{(\sqrt{3} - \sqrt{5})}{2}\right)
\][/tex]
Hence the rationalized form of [tex]\(\frac{s}{\sqrt{3} - \sqrt{5}}\)[/tex] is:
[tex]\[ \frac{s}{-\sqrt{5} + \sqrt{3}} \][/tex]
