Respuesta :
Given radical expression: [tex]\frac{3}{\sqrt{17}-\sqrt{2}}[/tex].
We need to find the fraction by which fraction will produce an equivalent fraction with a rational denominator.
In the given expression, we have [tex]\sqrt{17}-\sqrt{2}.[/tex]
Which is an irrational number in denominator.
In order to get rid irrational number from denominator and get a rational number in denominator, we need to multiply given expression by conjugate of the denominator.
The conjugate of [tex]\sqrt{17}-\sqrt{2}[/tex] is [tex]\sqrt{17}+\sqrt{2}[/tex].
Therefore, we need to multiply [tex]\frac{3}{\sqrt{17}-\sqrt{2}}[/tex] by [tex]\frac{\sqrt{17}+\sqrt{2}}{\sqrt{17}+\sqrt{2}}[/tex] fraction that will produce an equivalent fraction with a rational denominator.
Question asks to rationalize the denominator of the expression. Multiplying [tex]\dfrac{{\sqrt {17} + \sqrt 2}}{{\sqrt {17} + \sqrt 2}}[/tex] to the given expression will rationalize the denominator of the given expression.
The given expression is [tex]\dfrac{3}{\sqrt {17} - \sqrt 2}[/tex].
It is required to rationalize the denominator of the given expression.
To rationalize the denominator, multiply and divide the expression by [tex]\sqrt {17} + \sqrt 2[/tex] as,
[tex]\dfrac{3}{\sqrt {17} - \sqrt 2}\times \dfrac{\sqrt {17} + \sqrt 2}{\sqrt {17} + \sqrt 2}=\dfrac{3(\sqrt {17} + \sqrt 2)}{17-2}\\=\dfrac{3(\sqrt {17} + \sqrt 2)}{15}\\=\dfrac{\sqrt {17} + \sqrt 2}{5}\\=\dfrac{\sqrt {17} }{5}+\dfrac{ \sqrt 2}{5}[/tex]
Therefore, multiplying [tex]\dfrac{{\sqrt {17} + \sqrt 2}}{{\sqrt {17} + \sqrt 2}}[/tex] to the given expression will rationalize the denominator of the given expression.
For more details, refer to the link:
https://brainly.com/question/25292194
