Rationalize the denominator and simplify.
7-6sqrt 6/5+3sqrt6
To rationalize the denominator, we need to get rid of the square root in the denominator.
We can do this by multiplying both the numerator and the denominator by the conjugate of the denominator.
The conjugate of 5 + 3sqrt(6) is 5 - 3sqrt(6).
So, we have:
(7 - 6sqrt(6))/(5 + 3sqrt(6))
Multiplying the numerator and denominator by the conjugate:
((7 - 6sqrt(6))/(5 + 3sqrt(6))) * ((5 - 3sqrt(6))/(5 - 3sqrt(6)))
Simplifying the numerator:
(7 * 5) - (7 * 3sqrt(6)) - (6sqrt(6) * 5) + (6sqrt(6) * 3sqrt(6))
= 35 - 21sqrt(6) - 30sqrt(6) + 18 * sqrt(6)
= 35 - 51sqrt(6) + 18sqrt(6)
Simplifying the denominator:
(5 * 5) - (5 * 3sqrt(6)) + (3sqrt(6) * 5) - (3sqrt(6) * 3sqrt(6))
= 25 - 15sqrt(6) + 15sqrt(6) - 9 * sqrt(6)
= 25 - 9sqrt(6)
Putting it all together:
(35 - 51sqrt(6) + 18sqrt(6))/(25 - 9sqrt(6))
Simplifying further:
35 - 51sqrt(6) + 18sqrt(6)
= 35 - 33sqrt(6)
Thus, the simplified and rationalized expression is:
(35 - 33sqrt(6))/(25 - 9sqrt(6))