How do you rationalize the denominator and simplify #(15+3sqrt3)/(7+sqrt8)#?

Question

Ethan Adkins · Accepted Answer

See a solution process below:

To rationalize the denominator, multiply the fraction by #1# in the form of: #(7 - sqrt(8))/(7 - sqrt(8))#: #(7 - sqrt(8))/(7 - sqrt(8)) xx (15 + 3sqrt(3))/(7 + sqrt(8)) =># #((7 * 15) + (7 * 3sqrt(3)) - 15sqrt(8) - 3sqrt(8)sqrt(3))/((7 * 7) + 7sqrt(8) - 7sqrt(8) - (sqrt(8))^2) =># #(105 + 21sqrt(3) - 15sqrt(8) - 3sqrt(8 * 3))/(49 + (7sqrt(8) - 7sqrt(8)) - 8) =># #(105 + 21sqrt(3) - 15sqrt(8) - 3sqrt(24))/41 =># #(105 + 21sqrt(3) - 15sqrt(8) - 3sqrt(4 * 6))/41 =># #(105 + 21sqrt(3) - 15sqrt(8) - 3sqrt(4)sqrt(6))/41 =># #(105 + 21sqrt(3) - 15sqrt(8) - (3 * 2sqrt(6)))/41 =># #(105 + 21sqrt(3) - 15sqrt(8) - 6sqrt(6))/41#

Genesis Allen · Answer

undefined To rationalize the denominator and simplify the expression (15+3sqrt3)/(7+sqrt8), we can multiply both the numerator and denominator by the conjugate of the denominator.

The conjugate of 7+sqrt8 is 7-sqrt8.

By multiplying the numerator and denominator by 7-sqrt8, we get:

[(15+3sqrt3)(7-sqrt8)] / [(7+sqrt8)(7-sqrt8)]

Simplifying the numerator and denominator:

Numerator: (15+3sqrt3)(7-sqrt8) = 105 - 15sqrt8 + 21sqrt3 - 3sqrt3sqrt8
Denominator: (7+sqrt8)(7-sqrt8) = 49 - 8

Combining like terms:

Numerator: 105 - 15sqrt8 + 21sqrt3 - 3sqrt3sqrt8
Denominator: 49 - 8

Simplifying further:

Numerator: 97 - 15sqrt8 + 21sqrt3 - 3sqrt3sqrt8
Denominator: 41

Therefore, the simplified expression is:

(97 - 15sqrt8 + 21sqrt3 - 3sqrt3sqrt8) / 41