Rationalize each denominator. See Example 8.

3
————
4 + √5

Question

Rationalize each denominator. See Example 8.

3
————
 4 + √5

Accepted Answer

Welcome back. Everyone. In this problem, we want to rationalize the denominator of the radical expression 35 divided by nine plus the square root of 11. For our answer choices A is nine minus the square of 11 all divided by 70 B is nine minus the square of 11, all divided by two C is nine plus the squared of 11 all divided by 70 D is nine plus the square of all divided by two. Now, what do we know here? Well, we want to rationalize the denominator and the purpose of that is to make a caution involving a radical much simpler to interpret or simpler to read, to rationalize that denominator, we can find a conjugate of OURAA. Nor recall that to find a conjugate of a radical. So we call the, it means that we keep the same integer value but the opposite radical value. So that means that if I have nine plus the square of 11, then the conjugate after that would be equal to nine minus the square root of 11. OK. So now that I have that conjugate, then I can go ahead and multiply both the numerator and the denominator by that conjugate because when we do that multiplying the denominator will turn or, or get rid of our radical from the denominator. So that means I can rewrite divided by nine plus the square of 11 as 35 again, divided by nine plus the square of 11, multiplied by nine minus the square of 11 divided by nine minus the squared of 11. Now, when we multiply our denominator, it will turn out to be the difference of two squares. Again, recall that the difference of two squares states that if I'm multiplying A plus B by A minus B, then it's going to be the defense of two squares A squared minus B squared. So we can apply that here to our denominator because that means in our numerator where we have 35 multiplied by nine minus the squared of 11. And in our denominator, we will have nine squared minus the square root of 11 squared. Now, let's continue working 35. I'm not going to expand my numerator just yet. I'm going to work out what's going on in my denominator first. So nine squared is 81 and the square of 11 squared. Well, the square would cancel the square root and that would just leave us with and 81 minus 11 equals 70. Now, can I simplify my quotient any further? Well, 35 is a factor of 70. So I can go ahead and factor out 35 35 to 35 is 1, 35 goes into 72 times. And no, that leaves me with nine minus the square of 11 all divided by two. So that would have been my rationalized radical expression which when we go back to our answer choices would have been answer choice. B thanks a lot for watching everyone. I hope this video helped.

Rationalize each denominator. See Example 8. 3 ———— 4 + √5

Key Concepts

Rationalizing the Denominator

Conjugates

Simplifying Radicals

Watch next