How can you write the expression with a rationalized denominator ? (sqrt(3) - sqrt(6))/(sqrt(3) + sqrt(6))
To rationalize the denominator of the expression (sqrt(3) - sqrt(6))/(sqrt(3) + sqrt(6)), you need to multiply both the numerator and denominator by the conjugate of the denominator, which is (sqrt(3) - sqrt(6)).
Therefore, the rationalized form of the expression is:
((sqrt(3) - sqrt(6))/(sqrt(3) + sqrt(6))) * ((sqrt(3) - sqrt(6))/(sqrt(3) - sqrt(6)))
Expanding and simplifying the numerator and denominator gives:
= (sqrt(3) * sqrt(3) - sqrt(3) * sqrt(6) - sqrt(3) * sqrt(6) + sqrt(6) * sqrt(6)) / ((sqrt(3) * sqrt(3)) - (sqrt(6) * sqrt(6)))
= (3 - 2sqrt(18) + 6) / (3 - 6)
= (9 - 2sqrt(18)) / (-3)
= (9 - 2sqrt(3 * 6)) / (-3)
= (9 - 6sqrt(2 * 3)) / (-3)
= (9 - 6sqrt(6)) / (-3)
= -3 + 2sqrt(6)
Therefore, the expression (sqrt(3) - sqrt(6))/(sqrt(3) + sqrt(6)) can be written as -3 + 2sqrt(6) with a rationalized denominator.