How do you rationalize the denominator and simplify 6/(3+sqrt3)63+√3?

1 Answer
Noah G
Aug 30, 2016

The answer is 3 - sqrt(3)3−√3.

Explanation:

When rationalizing expressions where the denominators are binomials (two terms), we multiply the entire expression by the conjugate of the denominator.

The conjugate is meant to make a difference of squares when multiplied with its original expression. For example, the conjugate of a + b a+b is a -ba−b, since (a + b)(a - b) = a^2 + ab - ab - b^2 = a^2 - b^2(a+b)(a−b)=a2+ab−ab−b2=a2−b2, or a difference of squares.

The conjugate can always be found by switching the middle sign of the denominator. Hence, the conjugate of 3 + sqrt(3)3+√3 is 3 - sqrt(3)3−√3.

Let's start the rationalization process.

=6/(3 + sqrt(3)) xx (3 - sqrt(3))/(3 - sqrt(3))=63+√3×3−√33−√3

=(18 - 6sqrt(3))/(9 - 3sqrt(3) + 3sqrt(3) - sqrt(9))=18−6√39−3√3+3√3−√9

=(18 - 6sqrt(3))/(9 - 3)=18−6√39−3

=(6(3 - sqrt(3)))/(6)=6(3−√3)6

=3 - sqrt(3)=3−√3

Hopefully this helps!

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