The considered fraction is ationalized by multiplying the numerator and denominator by the expression as given by: Option A: √7 + 2
How to rationalize a fraction?
Suppose the given fraction is [tex]\dfrac{a}{b+c}[/tex]
Then the conjugate of the denominator is given by b - c
Thus, rationalizing the fraction will give us
[tex]\dfrac{a}{b+c} \times \dfrac{b-c}{b-c} = \dfrac{a(b-c)}{b^2 - c^2}[/tex]
(we used the identity [tex](b-c)(b+c) = b^2 - c^2[/tex])
We actually rationalize just for the use of that later denominator or numerator(if they seem to be helpful).
Remember that
[tex]\dfrac{b-c}{b-c} = 1[/tex]
thus, multiplying it with the fraction doesn't change its value, and just change the way how it looks. We assume that b-c is not 0
For this case, the given fraction is:
[tex]\dfrac{5}{\sqrt{7} - 2}[/tex]
Thus, here we will multiply the denominator with [tex]\sqrt{7} + 2[/tex] so that we can use the identity [tex](b-c)(b+c) = b^2 - c^2[/tex].
Thus, we get:
[tex]\dfrac{5}{\sqrt{7} - 2} \times \dfrac{\sqrt{7} + 2}{\sqrt{7} + 2} = \dfrac{5(\sqrt{7} + 2)}{(\sqrt{7})^2 - 2^2} = \dfrac{5(\sqrt{7} + 2)}{3}[/tex]
See, the denominator now looks a bit better. We try to make it look smooth and simplified because division is more complex than multiplication, so its better to have square root term in numerator than in denominator.
Thus, the considered fraction is ationalized by multiplying the numerator and denominator by the expression as given by: Option A: √7 + 2
Learn more about rationalization here;
https://brainly.com/question/14261303
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