How do you rationalize the denominator and simplify $(10-sqrt3)/(6+sqrt6)$ ?

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Answer 1 · 2016-03-15T18:59:25

$(60-10sqrt(6)-6sqrt(3)+3sqrt(2))/30$

$1$ . Start by multiplying the numerator and denominator by the conjugate of the fraction's denominator, $6-sqrt(6)$ .

$(10-sqrt(3))/(6+sqrt(6))$

$=(10-sqrt(3))/(6+sqrt(6))((6-sqrt(6))/(6-sqrt(6)))$

$2$ . Simplify the numerator.

$=(60-10sqrt(6)-6sqrt(3)+sqrt(18))/(6+sqrt(6))(1/(6-sqrt(6)))$

$=(60-10sqrt(6)-6sqrt(3)+3sqrt(2))/(6+sqrt(6))(1/(6-sqrt(6)))$

$3$ . Simplify the denominator. Note that it contains a difference of squares $(color(red)(a^2-b^2=(a+b)(a-b)))$ .

$=(60-10sqrt(6)-6sqrt(3)+3sqrt(2))/(36-6)$

$=color(green)(|bar(ul(color(white)(a/a)(60-10sqrt(6)-6sqrt(3)+3sqrt(2))/30color(white)(a/a)|)))$

How do you rationalize the denominator and simplify (10-sqrt3)/(6+sqrt6)(10-sqrt3)/(6+sqrt6)?