How To Solve: Rationalize Roots

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Hi All,

I have recently uploaded a video on YouTube to discuss Rationalize Roots in Detail:




Following is covered in the video

¤ How to Rationalize Roots
¤ Example 1 : Rationalize \(\frac{1}{√𝟑−√𝟐}\)
¤ Example 2 : Rationalize \(\frac{1}{√𝟒+√𝟑}\) + \(\frac{1}{√𝟑+√𝟐}\)

How to Rationalize Roots

¤ To Rationalize the denominator we do computations to move the root term to the numerator.
¤ This is usually done by multiplying the numerator and denominator with a conjugate of the denominator.
¤ Thus the denominator becomes a whole number.

Example 1 : Rationalize \(\frac{1}{√𝟑−√𝟐}\)

To Rationalize \(\frac{1}{√𝟑−√𝟐}\) we will multiply the numerator and the denominator with the conjugate of the denominator.
We can find the conjugate of √𝟑−√𝟐 by just inversing the sign between √𝟑 and √𝟐
=> Conjugate of √𝟑−√𝟐 will be √𝟑+√𝟐

=> \(\frac{1}{√𝟑−√𝟐}\) = \(\frac{1}{√𝟑−√𝟐}\) * \(\frac{√𝟑+√𝟐}{√𝟑+√𝟐}\)
= \(\frac{√𝟑+√𝟐}{(√𝟑−√𝟐) * (√𝟑+√𝟐)}\)

Now the denominator is of the form (a-b) * (a+b) and will be equal to \(a^2 - b^2\)

=> \(\frac{√𝟑+√𝟐}{(√𝟑−√𝟐) * (√𝟑+√𝟐)}\) = \(\frac{√𝟑+√𝟐}{(√𝟑)^2 − (√𝟐)^2}\) = \(\frac{√𝟑+√𝟐}{3 − 2}\) = √𝟑+√𝟐

Example 2 : Rationalize \(\frac{1}{√𝟒+√𝟑}\) + \(\frac{1}{√𝟑+√𝟐}\)

Following above logic we can find that \(\frac{1}{√𝟒+√𝟑}\) = √𝟒 - √𝟑 and \(\frac{1}{√𝟑+√𝟐}\) = √3 - √2

=> \(\frac{1}{√𝟒+√𝟑}\) + \(\frac{1}{√𝟑+√𝟐}\) = √𝟒 - √𝟑 + √3 - √2 = √𝟒 - √2 = 2 - √2

Hope it helps!
Good Luck!

Watch the following video to learn the Properties of Roots