Exercises 73–75 will help you prepare for the material covered in...

Question

Exercises 73–75 will help you prepare for the material covered in the next section.

Rationalize the denominator: (7 + 4√2)/(2 - 5√2).

Accepted Answer

Hello. Today we're going to be evaluating an expression by removing the radicals from the denominator. So the expression given to us is the quantity one plus three square root of three over the quantity nine minus two square root of three. Now, in order to clear the radical from the denominator we're going to need to multiply this by the conjugate of the denominator. Now the conjugate is multiplying by the opposite of the denominator. So we're going to use the same quantity but instead of having a minus we're going to have a plus. So the conjugate of the denominator is going to be nine plus two square root of 3/ plus two square root of three. Now we need to go ahead and multiply these quantities together and in order to do that we're going to need to foil both numerator and denominators together. So foiling the numerator together one times nine is going to give us 91 times two square root of three is going to give us two square root of three. Three square root of three times nine is going to give us square root of three and three square root of three times two square root of three is going to give us six times the square root of three squared And that's going to be our numerator for our denominator. We get nine times nine which is 81 nine times two square root of three which is 18 square root of three, negative two square root of three times nine which is negative, 18 square root of three and negative two squared of three times positive two square root of three, which is going to give us negative four to the square root of three squared. Now to go ahead and simplify the denominator positive 18 square root of three minus 18 square root of three is just gonna cancel each other out. And three the square root of three squared is just going to give us three. So we're left with the quantity four times 3 And that's going to give us 12 for the numerator, two square root of three plus 27 squared 23 is going to give us 29 Square root of three and just like the denominator, the square root of three squared is just going to give us three and six times three is going to give us 18. So simplifying this is going to give us nine plus 29 square root of three Plus 18/81 -12. And for the numerator we just have to go ahead and combine like terms we can combine 18 and nine and that's going to give us 27 And 81 -12 is going to give us 69. So what this quantity reduces to is plus 29 square root of 3/69 that's going to be the solution to this quantity with the radicals removed from the denominator. That also means that the solution to this problem is going to be a So I hope this video helps you in understanding how to remove the radical from the denominator, and I'll go ahead and see you all in the next video.

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