Question

Rationalize the denominator 3-sqrt(6)÷4+sqrt(5)

Answer 1

`(12-3√(5)-4√(6)+√(30))/(11)`

Explanation:

You have written 3 - ((√6)/4) + √5. The only denominator is 4, which is already rational. If you were to use appropriate grouping symbols (parentheses) we might expect to see ...

(3 -√6)/(4 +√5)

The denominator is rationalized by taking advantage of the factoring of the difference of squares:

a² - b² = (a -b)(a +b)

The denominator will be rational if we can square the value √5. By choosing a multiplier of (4-√5), we can do that:

`(3-√(6))/(4+√(5))=((3-√(6))(4-√(5)))/((4+√(5))(4-√(5)))\\\\=(3\cdot 4-3√(5)-4√(6)+√(30))/(4^2-(√(5))^2)=(12-3√(5)-4√(6)+√(30))/(11)`